[{"Name":"Introduction to Improper Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Improper Integrals - Type 1","Duration":"3m 35s","ChapterTopicVideoID":8815,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8815.jpeg","UploadDate":"2019-12-11T21:22:35.3930000","DurationForVideoObject":"PT3M35S","Description":null,"MetaTitle":"Improper Integrals - Type 1: Video + Workbook | Proprep","MetaDescription":"Improper Integrals - Introduction to Improper Integrals. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/improper-integrals/introduction-to-improper-integrals/vid8998","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this clip, we\u0027re going to learn about improper integrals."},{"Start":"00:04.110 ","End":"00:07.470","Text":"These are a kind of integral, but more generalized."},{"Start":"00:07.470 ","End":"00:10.470","Text":"They generalize what we previously thought were integrals."},{"Start":"00:10.470 ","End":"00:12.555","Text":"There are in fact 3 types,"},{"Start":"00:12.555 ","End":"00:14.910","Text":"and we\u0027ll learn to know all of them."},{"Start":"00:14.910 ","End":"00:16.980","Text":"But we start with first things first,"},{"Start":"00:16.980 ","End":"00:20.880","Text":"and that will be what we call type 1 improper integral."},{"Start":"00:20.880 ","End":"00:24.689","Text":"Now, a type 1 integral is like an integral,"},{"Start":"00:24.689 ","End":"00:29.129","Text":"but where at least 1 of the limits of integration is infinite."},{"Start":"00:29.129 ","End":"00:37.364","Text":"For example, the integral from a to infinity of f of x dx,"},{"Start":"00:37.364 ","End":"00:46.045","Text":"or we might have the integral from minus infinity to b of some function,"},{"Start":"00:46.045 ","End":"00:53.825","Text":"or we could even have from minus infinity to infinity of f of x dx."},{"Start":"00:53.825 ","End":"00:56.735","Text":"You agree with me that this is something new because"},{"Start":"00:56.735 ","End":"00:59.600","Text":"up until now we\u0027ve had 2 definite numbers,"},{"Start":"00:59.600 ","End":"01:00.920","Text":"a and b here,"},{"Start":"01:00.920 ","End":"01:04.940","Text":"and now we\u0027re allowing the possibility for infinity here or here,"},{"Start":"01:04.940 ","End":"01:06.830","Text":"or even in both places."},{"Start":"01:06.830 ","End":"01:08.795","Text":"Here are some examples."},{"Start":"01:08.795 ","End":"01:16.940","Text":"Let\u0027s take the integral from 1 to infinity of 1 over x squared dx."},{"Start":"01:16.940 ","End":"01:19.100","Text":"That would be an example of this kind."},{"Start":"01:19.100 ","End":"01:20.735","Text":"Let\u0027s take another example,"},{"Start":"01:20.735 ","End":"01:27.025","Text":"the integral of minus infinity to 4 of e^x dx."},{"Start":"01:27.025 ","End":"01:31.760","Text":"A third example corresponding to this 1 would be the integral from"},{"Start":"01:31.760 ","End":"01:38.765","Text":"minus infinity to infinity of 1 over 1 plus x squared dx."},{"Start":"01:38.765 ","End":"01:42.560","Text":"At this point, I\u0027d like to introduce some further conditions on the function"},{"Start":"01:42.560 ","End":"01:44.480","Text":"f. I didn\u0027t tell you before because I didn\u0027t"},{"Start":"01:44.480 ","End":"01:46.820","Text":"want to burden you with too many things at once."},{"Start":"01:46.820 ","End":"01:49.445","Text":"But there\u0027s 2 requirements we have of the function"},{"Start":"01:49.445 ","End":"01:53.449","Text":"f. The first condition that I need to impose on f,"},{"Start":"01:53.449 ","End":"01:56.825","Text":"is that I will have to require it to be continuous."},{"Start":"01:56.825 ","End":"02:01.400","Text":"Let\u0027s check that this indeed occurs in our 3 examples."},{"Start":"02:01.400 ","End":"02:03.170","Text":"1 over x squared, well,"},{"Start":"02:03.170 ","End":"02:06.349","Text":"you might say it\u0027s not continuous at x equals 0,"},{"Start":"02:06.349 ","End":"02:10.145","Text":"but 0 is outside our limits of integration."},{"Start":"02:10.145 ","End":"02:13.850","Text":"In between 1 and infinity, f is continuous,"},{"Start":"02:13.850 ","End":"02:18.120","Text":"and here e^x is continuous everywhere, so no problem."},{"Start":"02:18.120 ","End":"02:22.449","Text":"Also here, 1 over 1 plus x squared is continuous everywhere."},{"Start":"02:22.449 ","End":"02:26.900","Text":"Now, the second condition that we require off before we do an"},{"Start":"02:26.900 ","End":"02:31.580","Text":"improper integral is that f of x has to also be bounded."},{"Start":"02:31.580 ","End":"02:36.380","Text":"Now, bounded is a little bit more difficult to understand."},{"Start":"02:36.380 ","End":"02:39.530","Text":"In practice, in all the examples you\u0027ll probably encounter,"},{"Start":"02:39.530 ","End":"02:45.755","Text":"it will be that f doesn\u0027t tend to plus or minus infinity."},{"Start":"02:45.755 ","End":"02:49.850","Text":"Theoretically, what it means is that there are 2 numbers, let\u0027s say,"},{"Start":"02:49.850 ","End":"02:52.835","Text":"a little m and big m,"},{"Start":"02:52.835 ","End":"02:55.700","Text":"where the function is always between these 2 numbers."},{"Start":"02:55.700 ","End":"02:57.875","Text":"That\u0027s strictly speaking, what bounded means."},{"Start":"02:57.875 ","End":"02:59.000","Text":"Though, as I say, in practice,"},{"Start":"02:59.000 ","End":"03:01.490","Text":"you just have to think of it as not going to infinity."},{"Start":"03:01.490 ","End":"03:05.180","Text":"For example, here this number is bounded between 1 and 0."},{"Start":"03:05.180 ","End":"03:08.525","Text":"1 over x squared is always between 1 and 0 in this range."},{"Start":"03:08.525 ","End":"03:12.710","Text":"This 1 is also bounded between 0 and e^4,"},{"Start":"03:12.710 ","End":"03:17.655","Text":"and this 1 is always less than 1 and greater than 0."},{"Start":"03:17.655 ","End":"03:19.910","Text":"They\u0027re actually bounded. But in practice,"},{"Start":"03:19.910 ","End":"03:22.655","Text":"as I say, this is the condition that we\u0027ll be using."},{"Start":"03:22.655 ","End":"03:24.950","Text":"But we don\u0027t usually go back to these conditions,"},{"Start":"03:24.950 ","End":"03:27.800","Text":"but you should remember them somewhere in the background."},{"Start":"03:27.800 ","End":"03:29.795","Text":"Now the big question is,"},{"Start":"03:29.795 ","End":"03:33.274","Text":"how do we compute this new kind of integral?"},{"Start":"03:33.274 ","End":"03:35.760","Text":"That\u0027s what I\u0027m going to show you now."}],"ID":8998},{"Watched":false,"Name":"Improper Integrals - Type 1 - Example 1","Duration":"4m 2s","ChapterTopicVideoID":8810,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.600","Text":"To show you how to solve an improper integral,"},{"Start":"00:02.600 ","End":"00:04.515","Text":"it\u0027s best to start with an example."},{"Start":"00:04.515 ","End":"00:13.980","Text":"So let\u0027s take the example of the integral from 1 to infinity of 1 over x squared dx."},{"Start":"00:13.980 ","End":"00:17.550","Text":"Now the question is, what to do with this infinity symbol?"},{"Start":"00:17.550 ","End":"00:19.590","Text":"Well, here\u0027s where we get around that."},{"Start":"00:19.590 ","End":"00:24.750","Text":"We first of all start by taking the integral from 1 to infinity,"},{"Start":"00:24.750 ","End":"00:26.325","Text":"but to some large number,"},{"Start":"00:26.325 ","End":"00:29.715","Text":"we\u0027ll choose a variable b, x squared dx,"},{"Start":"00:29.715 ","End":"00:34.725","Text":"and then take the limit as b goes to infinity."},{"Start":"00:34.725 ","End":"00:36.690","Text":"This is like putting infinity here."},{"Start":"00:36.690 ","End":"00:39.450","Text":"We put a finite number and let it go to infinity."},{"Start":"00:39.450 ","End":"00:41.925","Text":"Let\u0027s see what this works out to."},{"Start":"00:41.925 ","End":"00:43.890","Text":"The limit stays the limit,"},{"Start":"00:43.890 ","End":"00:45.950","Text":"b goes to infinity."},{"Start":"00:45.950 ","End":"00:51.309","Text":"The indefinite integral of 1 over x squared is minus 1 over x,"},{"Start":"00:51.309 ","End":"00:56.180","Text":"and this we have to take between the limits b and 1,"},{"Start":"00:56.180 ","End":"00:57.500","Text":"which means we substitute this,"},{"Start":"00:57.500 ","End":"01:03.815","Text":"substitute this and subtract what we get is the limit as b goes to infinity."},{"Start":"01:03.815 ","End":"01:05.945","Text":"Now if I put in 1,"},{"Start":"01:05.945 ","End":"01:09.260","Text":"it\u0027s minus 1 over 1, which is minus 1,"},{"Start":"01:09.260 ","End":"01:11.435","Text":"but I\u0027m subtracting it so it\u0027s 1,"},{"Start":"01:11.435 ","End":"01:15.445","Text":"and if I put in b, it\u0027s minus 1 over b."},{"Start":"01:15.445 ","End":"01:18.740","Text":"What we have is the limit as b goes to infinity minus"},{"Start":"01:18.740 ","End":"01:22.195","Text":"1 over b and that of course equals 1."},{"Start":"01:22.195 ","End":"01:24.940","Text":"Now another definition which we\u0027ll repeat later,"},{"Start":"01:24.940 ","End":"01:26.690","Text":"but in this example,"},{"Start":"01:26.690 ","End":"01:29.615","Text":"what we got was a finite number,"},{"Start":"01:29.615 ","End":"01:32.570","Text":"an actual limit, not something infinite."},{"Start":"01:32.570 ","End":"01:37.220","Text":"In this case, what we say is that the integral converges."},{"Start":"01:37.220 ","End":"01:41.490","Text":"I would write the integral converges to 1."},{"Start":"01:41.490 ","End":"01:43.395","Text":"It\u0027s the same 1."},{"Start":"01:43.395 ","End":"01:46.365","Text":"I\u0027ll emphasize the word converges."},{"Start":"01:46.365 ","End":"01:48.515","Text":"We\u0027ll come across it again later."},{"Start":"01:48.515 ","End":"01:51.545","Text":"Basically done except for those who have studied"},{"Start":"01:51.545 ","End":"01:54.740","Text":"areas and the definite integral as an area,"},{"Start":"01:54.740 ","End":"01:58.145","Text":"I\u0027d like to show you the geometrical meaning of this."},{"Start":"01:58.145 ","End":"01:59.825","Text":"Let me start with a quick sketch."},{"Start":"01:59.825 ","End":"02:03.920","Text":"Let\u0027s take y-axis and we\u0027ll take x-axis."},{"Start":"02:03.920 ","End":"02:06.020","Text":"We\u0027ll draw 1 over x squared,"},{"Start":"02:06.020 ","End":"02:09.740","Text":"which is something like this and it has asymptotes,"},{"Start":"02:09.740 ","End":"02:13.460","Text":"goes up to infinity here and it goes to 0 here."},{"Start":"02:13.460 ","End":"02:19.085","Text":"What this integral represents is let\u0027s say that 1 is here and from 1 to infinity,"},{"Start":"02:19.085 ","End":"02:21.920","Text":"the definite integral is the area under the curve."},{"Start":"02:21.920 ","End":"02:23.990","Text":"So what we have is the area under the curve,"},{"Start":"02:23.990 ","End":"02:27.725","Text":"it\u0027s bounded above and below by the graph and the axis,"},{"Start":"02:27.725 ","End":"02:31.775","Text":"and on the left by the line where x equals 1."},{"Start":"02:31.775 ","End":"02:36.920","Text":"But there is no boundary on the right because this actually never touches the axis."},{"Start":"02:36.920 ","End":"02:39.425","Text":"It approaches asymptotically, but never touches."},{"Start":"02:39.425 ","End":"02:44.765","Text":"This area is what we are looking for in this integral and so what this means,"},{"Start":"02:44.765 ","End":"02:46.295","Text":"because the integral is 1,"},{"Start":"02:46.295 ","End":"02:47.600","Text":"is that this whole area,"},{"Start":"02:47.600 ","End":"02:50.375","Text":"even though it has an infinite length,"},{"Start":"02:50.375 ","End":"02:52.790","Text":"is actually finite in area."},{"Start":"02:52.790 ","End":"02:54.635","Text":"So this area, let\u0027s call it a,"},{"Start":"02:54.635 ","End":"02:56.410","Text":"is equal to 1,"},{"Start":"02:56.410 ","End":"02:59.270","Text":"and that\u0027s the geometrical significance of this."},{"Start":"02:59.270 ","End":"03:02.525","Text":"So this was the graph y equals 1 over x squared,"},{"Start":"03:02.525 ","End":"03:04.955","Text":"the y-axis, the x-axis,"},{"Start":"03:04.955 ","End":"03:07.790","Text":"the area, even though it\u0027s infinite length."},{"Start":"03:07.790 ","End":"03:11.495","Text":"I\u0027d like to generalize from the example we just did and say that"},{"Start":"03:11.495 ","End":"03:15.950","Text":"the integral from a to infinity of f of"},{"Start":"03:15.950 ","End":"03:21.260","Text":"x dx is defined as the limit as b goes to"},{"Start":"03:21.260 ","End":"03:28.039","Text":"infinity of the integral from a to b of f of x dx."},{"Start":"03:28.039 ","End":"03:33.800","Text":"But provided that this limit exists and is finite,"},{"Start":"03:33.800 ","End":"03:36.409","Text":"if this limit exists and is finite,"},{"Start":"03:36.409 ","End":"03:40.450","Text":"then we also say that this integral converges."},{"Start":"03:40.450 ","End":"03:42.185","Text":"If this is not the case,"},{"Start":"03:42.185 ","End":"03:47.120","Text":"then we say that the integral diverges otherwise"},{"Start":"03:47.120 ","End":"03:52.910","Text":"means that either the limit doesn\u0027t exist or it sort of exists but is infinite."},{"Start":"03:52.910 ","End":"03:54.544","Text":"So in the other cases,"},{"Start":"03:54.544 ","End":"03:58.475","Text":"we say that it\u0027s diverges and converges."},{"Start":"03:58.475 ","End":"04:03.540","Text":"I think I\u0027ll show you next an example of a divergent integral."}],"ID":8999},{"Watched":false,"Name":"Improper Integrals - Type 1 - Example 2","Duration":"2m 17s","ChapterTopicVideoID":8811,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.275","Text":"This time I\u0027d like to show you an example of a divergent interval."},{"Start":"00:04.275 ","End":"00:06.630","Text":"It\u0027s going to be similar to the 1 we had last time,"},{"Start":"00:06.630 ","End":"00:08.475","Text":"but the last 1 was convergent."},{"Start":"00:08.475 ","End":"00:11.310","Text":"It\u0027s again going to be from 1 to infinity."},{"Start":"00:11.310 ","End":"00:14.139","Text":"But last time, we had 1 over x squared,"},{"Start":"00:14.139 ","End":"00:17.250","Text":"and this time, we\u0027ll take 1 over x so that\u0027s different."},{"Start":"00:17.250 ","End":"00:20.415","Text":"The same technique we use is to take,"},{"Start":"00:20.415 ","End":"00:23.250","Text":"first of all, the integral from 1 set of infinity."},{"Start":"00:23.250 ","End":"00:26.900","Text":"We take it up to b of 1 over x dx,"},{"Start":"00:26.900 ","End":"00:30.230","Text":"and then we let b go to infinity with the limit,"},{"Start":"00:30.230 ","End":"00:32.210","Text":"b goes to infinity."},{"Start":"00:32.210 ","End":"00:35.375","Text":"What we do is we take an indefinite integral, in this case,"},{"Start":"00:35.375 ","End":"00:38.840","Text":"natural log of absolute value of x,"},{"Start":"00:38.840 ","End":"00:40.160","Text":"but we\u0027re on the positive,"},{"Start":"00:40.160 ","End":"00:41.900","Text":"so natural log of x."},{"Start":"00:41.900 ","End":"00:48.000","Text":"We need to take this between the limits b and 1,"},{"Start":"00:48.000 ","End":"00:52.729","Text":"limit as b goes to infinity."},{"Start":"00:52.729 ","End":"00:54.620","Text":"What we do is we substitute b,"},{"Start":"00:54.620 ","End":"00:58.490","Text":"substitute 1 and subtract so this is equal to"},{"Start":"00:58.490 ","End":"01:05.870","Text":"the limit as b goes to infinity of natural log of b,"},{"Start":"01:05.870 ","End":"01:08.555","Text":"minus natural log of 1."},{"Start":"01:08.555 ","End":"01:11.735","Text":"The natural log of 1 is 0,"},{"Start":"01:11.735 ","End":"01:15.035","Text":"but it doesn\u0027t really matter whatever finite number it is,"},{"Start":"01:15.035 ","End":"01:17.015","Text":"because natural log of b,"},{"Start":"01:17.015 ","End":"01:19.715","Text":"when b goes to infinity is infinity,"},{"Start":"01:19.715 ","End":"01:22.400","Text":"the natural logarithm function tends to infinity."},{"Start":"01:22.400 ","End":"01:24.035","Text":"It gets larger and larger."},{"Start":"01:24.035 ","End":"01:31.415","Text":"Basically, this is equal to plus infinity and so we say that it is divergent."},{"Start":"01:31.415 ","End":"01:34.520","Text":"Or sometimes, we say it diverges to infinity."},{"Start":"01:34.520 ","End":"01:37.135","Text":"In any event, it\u0027s not convergent."},{"Start":"01:37.135 ","End":"01:40.850","Text":"Those who study the definite integral as an area,"},{"Start":"01:40.850 ","End":"01:42.650","Text":"I\u0027ll just illustrate this."},{"Start":"01:42.650 ","End":"01:45.620","Text":"In contrast to the previous exercise,"},{"Start":"01:45.620 ","End":"01:47.959","Text":"we had a very similar picture."},{"Start":"01:47.959 ","End":"01:50.600","Text":"We had a graph of 1 over x squared."},{"Start":"01:50.600 ","End":"01:55.880","Text":"Here we have the graph of 1 over x and also from x equals 1."},{"Start":"01:55.880 ","End":"01:59.870","Text":"We have this area and it makes a real big difference."},{"Start":"01:59.870 ","End":"02:01.520","Text":"When it was 1 over x squared,"},{"Start":"02:01.520 ","End":"02:02.960","Text":"we got something convergent."},{"Start":"02:02.960 ","End":"02:05.390","Text":"We had infinite length but a finite area."},{"Start":"02:05.390 ","End":"02:07.610","Text":"This time, it\u0027s very similar,"},{"Start":"02:07.610 ","End":"02:11.090","Text":"but it must be a bit larger or a lot larger because time,"},{"Start":"02:11.090 ","End":"02:14.360","Text":"the area here turned out not to be finite,"},{"Start":"02:14.360 ","End":"02:18.090","Text":"but was infinite, so much for this example."}],"ID":9000},{"Watched":false,"Name":"Improper Integrals - Type 1 - Example 3","Duration":"1m 55s","ChapterTopicVideoID":8812,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.235","Text":"Now I want to give another example."},{"Start":"00:02.235 ","End":"00:05.730","Text":"We haven\u0027t done one with the lower integration limit is infinite,"},{"Start":"00:05.730 ","End":"00:09.180","Text":"so we\u0027ll take something from minus infinity here to 0."},{"Start":"00:09.180 ","End":"00:12.630","Text":"This will be e to the x dx."},{"Start":"00:12.630 ","End":"00:16.305","Text":"Very similar to what we did with infinity here, same idea."},{"Start":"00:16.305 ","End":"00:21.990","Text":"We take the limit and then we take the integral from 0 down to a,"},{"Start":"00:21.990 ","End":"00:23.070","Text":"or from a to 0,"},{"Start":"00:23.070 ","End":"00:26.250","Text":"but this time we let a go to minus infinity"},{"Start":"00:26.250 ","End":"00:29.250","Text":"and the same e to the power x dx."},{"Start":"00:29.250 ","End":"00:32.580","Text":"What we do is we take the indefinite integral here,"},{"Start":"00:32.580 ","End":"00:34.645","Text":"which is also e to the x,"},{"Start":"00:34.645 ","End":"00:38.690","Text":"place it between a and 0 means we substitute this,"},{"Start":"00:38.690 ","End":"00:40.400","Text":"substitute this and subtract,"},{"Start":"00:40.400 ","End":"00:46.010","Text":"but we still have to have the limit as a goes to minus infinity."},{"Start":"00:46.010 ","End":"00:49.985","Text":"e to the power 0 is 1, e to the power a, is e to the a,"},{"Start":"00:49.985 ","End":"00:58.745","Text":"so we have 1 minus e to the a and the limit as a goes to minus infinity."},{"Start":"00:58.745 ","End":"01:01.310","Text":"Now e to the minus infinity is 0,"},{"Start":"01:01.310 ","End":"01:04.265","Text":"so this is just equal to 1."},{"Start":"01:04.265 ","End":"01:07.570","Text":"I can write that the integral converges to 1,"},{"Start":"01:07.570 ","End":"01:10.340","Text":"but sometimes we just say that the integral is convergent"},{"Start":"01:10.340 ","End":"01:12.605","Text":"and don\u0027t specify the actual limit."},{"Start":"01:12.605 ","End":"01:16.820","Text":"A little sketch for those who know about areas and integrals."},{"Start":"01:16.820 ","End":"01:21.070","Text":"Take a y-axis, an x-axis,"},{"Start":"01:21.070 ","End":"01:24.020","Text":"e to the power x is something like this."},{"Start":"01:24.020 ","End":"01:26.600","Text":"It goes through the point 1."},{"Start":"01:26.600 ","End":"01:28.700","Text":"What we are saying here,"},{"Start":"01:28.700 ","End":"01:34.310","Text":"when we take from minus infinity to 0 is here\u0027s the 0 for x"},{"Start":"01:34.310 ","End":"01:39.200","Text":"and here is the limit between the curve and the x-axis"},{"Start":"01:39.200 ","End":"01:43.070","Text":"and it isn\u0027t bounded on the left because this is asymptotic,"},{"Start":"01:43.070 ","End":"01:45.410","Text":"it goes to 0 but never actually reaches there."},{"Start":"01:45.410 ","End":"01:47.435","Text":"What we\u0027re saying is that this area,"},{"Start":"01:47.435 ","End":"01:48.920","Text":"even though it\u0027s infinite in length,"},{"Start":"01:48.920 ","End":"01:53.435","Text":"it\u0027s finite area and this area is equal to 1."},{"Start":"01:53.435 ","End":"01:55.920","Text":"That\u0027s this example."}],"ID":9001},{"Watched":false,"Name":"Improper Integrals - Type 1 - Example 4","Duration":"4m 38s","ChapterTopicVideoID":8813,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"This time I want to give an example where both the limits of integration are infinite."},{"Start":"00:04.650 ","End":"00:07.980","Text":"What I mean is integral from minus infinity to infinity."},{"Start":"00:07.980 ","End":"00:13.605","Text":"The function I\u0027ll use for an example is 1 over 1 plus x squared dx."},{"Start":"00:13.605 ","End":"00:17.250","Text":"The way to tackle this kind of a problem is just to break it up into 2."},{"Start":"00:17.250 ","End":"00:18.239","Text":"Pick any number."},{"Start":"00:18.239 ","End":"00:20.325","Text":"I\u0027m going to choose 0."},{"Start":"00:20.325 ","End":"00:23.340","Text":"Instead of the integral from minus infinity to infinity,"},{"Start":"00:23.340 ","End":"00:29.415","Text":"we take from minus infinity to 0 and then from 0 to infinity."},{"Start":"00:29.415 ","End":"00:30.720","Text":"It\u0027s the same thing,"},{"Start":"00:30.720 ","End":"00:35.060","Text":"1 over 1 plus x squared dx and here,"},{"Start":"00:35.060 ","End":"00:38.600","Text":"1 over 1 plus x squared dx."},{"Start":"00:38.600 ","End":"00:40.250","Text":"The 0 is arbitrary."},{"Start":"00:40.250 ","End":"00:43.670","Text":"I could have chosen minus infinity to 4 and 4 to infinity,"},{"Start":"00:43.670 ","End":"00:48.035","Text":"but I\u0027m guessing that 0 will be convenient for substitution and so on."},{"Start":"00:48.035 ","End":"00:51.200","Text":"You break it up into 2, and there\u0027s a plus here, of course."},{"Start":"00:51.200 ","End":"00:54.470","Text":"What I\u0027m going to do is compute each one separately and then add."},{"Start":"00:54.470 ","End":"00:59.870","Text":"I\u0027ll call this one asterisk and I\u0027ll call this one, double asterisk."},{"Start":"00:59.870 ","End":"01:02.120","Text":"Let\u0027s go first with the asterisk."},{"Start":"01:02.120 ","End":"01:03.620","Text":"We\u0027ve learned how to do this,"},{"Start":"01:03.620 ","End":"01:07.249","Text":"so we take minus infinity as a variable."},{"Start":"01:07.249 ","End":"01:15.670","Text":"I take the integral from a to 0 of 1 over 1 plus x squared dx."},{"Start":"01:15.670 ","End":"01:16.970","Text":"To evaluate this limit,"},{"Start":"01:16.970 ","End":"01:18.785","Text":"we need an indefinite integral."},{"Start":"01:18.785 ","End":"01:24.050","Text":"You might remember that the indefinite integral of this expression is the arctangent."},{"Start":"01:24.050 ","End":"01:30.155","Text":"What we have is the limit of the arctangent of x."},{"Start":"01:30.155 ","End":"01:34.700","Text":"This arctangent is taken between 0 and a,"},{"Start":"01:34.700 ","End":"01:38.150","Text":"which means substitute 0 and substitute a and subtract,"},{"Start":"01:38.150 ","End":"01:41.644","Text":"and the limit of course, is a goes to minus infinity."},{"Start":"01:41.644 ","End":"01:43.320","Text":"This is equal to,"},{"Start":"01:43.320 ","End":"01:46.519","Text":"the arctangent of 0 is 0."},{"Start":"01:46.519 ","End":"01:52.084","Text":"On the calculator you can do shift and then tangent to get the arctangent."},{"Start":"01:52.084 ","End":"01:53.959","Text":"Arctangent of 0 is 0,"},{"Start":"01:53.959 ","End":"01:58.160","Text":"so we\u0027re just left with minus arctangent a."},{"Start":"01:58.160 ","End":"02:01.355","Text":"Then I want to let a go to infinity."},{"Start":"02:01.355 ","End":"02:02.615","Text":"I\u0027ll just write it outside."},{"Start":"02:02.615 ","End":"02:06.440","Text":"The limit as a goes to minus infinity."},{"Start":"02:06.440 ","End":"02:12.740","Text":"This is equal to just minus arctangent of minus infinity."},{"Start":"02:12.740 ","End":"02:14.330","Text":"If any of you remember,"},{"Start":"02:14.330 ","End":"02:19.720","Text":"the arctangent of minus infinity is minus Pi over 2."},{"Start":"02:19.720 ","End":"02:23.290","Text":"Altogether the minus with the minus becomes plus,"},{"Start":"02:23.290 ","End":"02:28.635","Text":"so this integral is equal to Pi over 2."},{"Start":"02:28.635 ","End":"02:32.060","Text":"At this stage, I want to mention something important that I didn\u0027t say before."},{"Start":"02:32.060 ","End":"02:34.960","Text":"When we split an integral up into 2 parts,"},{"Start":"02:34.960 ","End":"02:36.710","Text":"if the first one doesn\u0027t converge,"},{"Start":"02:36.710 ","End":"02:37.960","Text":"we can stop already."},{"Start":"02:37.960 ","End":"02:40.220","Text":"This time it did, so we have to check the second one."},{"Start":"02:40.220 ","End":"02:42.410","Text":"But if the second one for example, doesn\u0027t converge,"},{"Start":"02:42.410 ","End":"02:44.060","Text":"then we don\u0027t have an integral."},{"Start":"02:44.060 ","End":"02:47.930","Text":"Just to remind you, converge means that it has a finite limit."},{"Start":"02:47.930 ","End":"02:50.060","Text":"The integral not only exists,"},{"Start":"02:50.060 ","End":"02:52.910","Text":"the limit, that is, exists but is finite."},{"Start":"02:52.910 ","End":"02:56.150","Text":"Let\u0027s get onto the double asterisk."},{"Start":"02:56.150 ","End":"03:06.440","Text":"This time b goes to infinity of the integral from 0 to b of 1 over 1 plus x squared dx."},{"Start":"03:06.440 ","End":"03:14.190","Text":"What we get here is also the limit of the arctangent of x,"},{"Start":"03:14.190 ","End":"03:16.620","Text":"this time from 0 to b,"},{"Start":"03:16.620 ","End":"03:18.860","Text":"b goes to infinity."},{"Start":"03:18.860 ","End":"03:24.930","Text":"This gives us the limit of arctangent b minus arctangent 0,"},{"Start":"03:24.930 ","End":"03:26.595","Text":"b goes to infinity."},{"Start":"03:26.595 ","End":"03:30.525","Text":"As we mentioned before the arctangent of 0 is 0."},{"Start":"03:30.525 ","End":"03:35.150","Text":"We basically have the limit as b goes to infinity of arctangent b,"},{"Start":"03:35.150 ","End":"03:37.265","Text":"and that\u0027s arctangent infinity."},{"Start":"03:37.265 ","End":"03:41.270","Text":"Just like the arctangent of minus infinity was minus Pi over 2,"},{"Start":"03:41.270 ","End":"03:45.570","Text":"it turns out that the arctangent of infinity is Pi over 2."},{"Start":"03:45.570 ","End":"03:49.175","Text":"Now if we look back at where we split it up,"},{"Start":"03:49.175 ","End":"03:51.670","Text":"asterisk comes out Pi over 2."},{"Start":"03:51.670 ","End":"03:53.550","Text":"It also came out to Pi over 2,"},{"Start":"03:53.550 ","End":"03:55.515","Text":"so these both converge."},{"Start":"03:55.515 ","End":"03:58.460","Text":"We can say that this integral is the sum of those 2,"},{"Start":"03:58.460 ","End":"04:05.375","Text":"asterisk plus double asterisk is equal to Pi over 2 plus Pi over 2."},{"Start":"04:05.375 ","End":"04:08.295","Text":"Altogether we get that this is Pi."},{"Start":"04:08.295 ","End":"04:11.665","Text":"The answer to our question is Pi."},{"Start":"04:11.665 ","End":"04:15.230","Text":"What this shows us is that the area altogether,"},{"Start":"04:15.230 ","End":"04:16.745","Text":"even though it\u0027s infinite,"},{"Start":"04:16.745 ","End":"04:18.860","Text":"is equal to Pi over 2."},{"Start":"04:18.860 ","End":"04:22.835","Text":"In fact we even showed more that, if we take the 2 bits separately,"},{"Start":"04:22.835 ","End":"04:26.985","Text":"this is Pi over 2 and this is Pi over 2,"},{"Start":"04:26.985 ","End":"04:32.670","Text":"and so altogether we got that Pi over 2 plus Pi over 2 was equal to Pi for the area."},{"Start":"04:32.670 ","End":"04:33.750","Text":"That was also the improper integral."},{"Start":"04:33.750 ","End":"04:39.340","Text":"That\u0027s another example and onto the last one."}],"ID":9002},{"Watched":false,"Name":"Improper Integrals - Type 1 - Example 5","Duration":"6m 23s","ChapterTopicVideoID":8814,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"Now our last example in the Type 1 improper integral,"},{"Start":"00:04.140 ","End":"00:08.295","Text":"let\u0027s take another one of those that goes from minus infinity to infinity,"},{"Start":"00:08.295 ","End":"00:10.935","Text":"and this time it will be a little more complicated."},{"Start":"00:10.935 ","End":"00:16.320","Text":"Minus 2x e to the power of minus x squared dx."},{"Start":"00:16.320 ","End":"00:19.830","Text":"Our usual technique is to break it up into two integrals."},{"Start":"00:19.830 ","End":"00:22.260","Text":"We choose a finite number in the middle,"},{"Start":"00:22.260 ","End":"00:25.080","Text":"and we often choose zero because it\u0027s easy to work with."},{"Start":"00:25.080 ","End":"00:32.144","Text":"We make this the integral from minus infinity to 0 plus another integral"},{"Start":"00:32.144 ","End":"00:40.395","Text":"from 0 to infinity and the same thing inside minus 2x e^minus x squared dx,"},{"Start":"00:40.395 ","End":"00:45.020","Text":"minus 2x^minus x squared dx."},{"Start":"00:45.020 ","End":"00:52.160","Text":"Let\u0027s say the first one we\u0027ll call asterisk and the second one double asterisk."},{"Start":"00:52.160 ","End":"00:54.815","Text":"We\u0027ll figure each one out separately and add."},{"Start":"00:54.815 ","End":"00:59.510","Text":"But remember, if any one of these two diverges,"},{"Start":"00:59.510 ","End":"01:02.825","Text":"then the whole thing is divergent and we don\u0027t have an integral."},{"Start":"01:02.825 ","End":"01:04.765","Text":"Well, let\u0027s start out with the first."},{"Start":"01:04.765 ","End":"01:06.180","Text":"If it fails, we can stop,"},{"Start":"01:06.180 ","End":"01:07.279","Text":"but if it succeeds,"},{"Start":"01:07.279 ","End":"01:09.410","Text":"if it diverges, we have to continue."},{"Start":"01:09.410 ","End":"01:19.740","Text":"The integral from minus infinity to 0 of minus 2x e^minus x squared dx is equal to,"},{"Start":"01:19.740 ","End":"01:23.570","Text":"I will replace this limit below with any variable, say a,"},{"Start":"01:23.570 ","End":"01:28.340","Text":"and then take the limit of that as it goes to minus infinity."},{"Start":"01:28.340 ","End":"01:30.590","Text":"Now instead of the minus infinity,"},{"Start":"01:30.590 ","End":"01:34.345","Text":"I put a to 0 minus 2x,"},{"Start":"01:34.345 ","End":"01:37.445","Text":"e^minus x squared dx."},{"Start":"01:37.445 ","End":"01:40.699","Text":"There is a little formula that you might recall."},{"Start":"01:40.699 ","End":"01:45.620","Text":"If I have e to the power of a function and alongside it the derivative of that function,"},{"Start":"01:45.620 ","End":"01:50.224","Text":"the indefinite integral is e to the power of that function."},{"Start":"01:50.224 ","End":"01:51.710","Text":"C doesn\u0027t really matter here,"},{"Start":"01:51.710 ","End":"01:53.630","Text":"if you don\u0027t remember it or you\u0027re not convinced,"},{"Start":"01:53.630 ","End":"01:55.885","Text":"just try differentiating e^f."},{"Start":"01:55.885 ","End":"01:58.315","Text":"The derivative of e^f is just e^f,"},{"Start":"01:58.315 ","End":"02:00.680","Text":"but we also have to multiply by the inner derivative,"},{"Start":"02:00.680 ","End":"02:02.090","Text":"which is f prime."},{"Start":"02:02.090 ","End":"02:03.740","Text":"Now how does this help me here?"},{"Start":"02:03.740 ","End":"02:08.360","Text":"What I would like to do is to take the minus x"},{"Start":"02:08.360 ","End":"02:14.945","Text":"squared as my function f. Its derivative is minus 2x, is already here."},{"Start":"02:14.945 ","End":"02:16.969","Text":"According to this formula,"},{"Start":"02:16.969 ","End":"02:20.630","Text":"this will equal the indefinite integral of this will just"},{"Start":"02:20.630 ","End":"02:24.319","Text":"be e to the power of minus x squared."},{"Start":"02:24.319 ","End":"02:26.255","Text":"Now, what do I get?"},{"Start":"02:26.255 ","End":"02:31.570","Text":"I still have to take the limit as a goes to minus infinity,"},{"Start":"02:31.570 ","End":"02:35.195","Text":"and what I have is this thing which I must evaluate"},{"Start":"02:35.195 ","End":"02:38.900","Text":"between a and 0 means substitute this,"},{"Start":"02:38.900 ","End":"02:40.460","Text":"substitute this and subtract."},{"Start":"02:40.460 ","End":"02:42.065","Text":"This is the limit."},{"Start":"02:42.065 ","End":"02:43.835","Text":"Now if I put in 0,"},{"Start":"02:43.835 ","End":"02:46.510","Text":"I get e^0 which is 1,"},{"Start":"02:46.510 ","End":"02:48.210","Text":"if I put in a,"},{"Start":"02:48.210 ","End":"02:52.475","Text":"I get minus e^minus a squared."},{"Start":"02:52.475 ","End":"02:56.465","Text":"A goes to minus infinity of this."},{"Start":"02:56.465 ","End":"02:59.840","Text":"Now here we can just substitute a is minus infinity,"},{"Start":"02:59.840 ","End":"03:02.329","Text":"minus infinity squared is infinity."},{"Start":"03:02.329 ","End":"03:06.470","Text":"Again, minus infinity, e^minus infinity is 0."},{"Start":"03:06.470 ","End":"03:09.140","Text":"Altogether, we\u0027re just left with 1."},{"Start":"03:09.140 ","End":"03:12.050","Text":"Asterisk came out to be equal to 1,"},{"Start":"03:12.050 ","End":"03:16.200","Text":"and now we have to check the second one because this one converge to 1."},{"Start":"03:16.200 ","End":"03:19.040","Text":"Let\u0027s check the double asterisk,"},{"Start":"03:19.040 ","End":"03:26.945","Text":"the integral from 0 to infinity of minus 2x e^minus x squared dx."},{"Start":"03:26.945 ","End":"03:28.340","Text":"Pretty much like here,"},{"Start":"03:28.340 ","End":"03:31.310","Text":"except that this time the infinity is on the top,"},{"Start":"03:31.310 ","End":"03:33.715","Text":"so we\u0027ll take it as the limit."},{"Start":"03:33.715 ","End":"03:36.140","Text":"The one on the top I like to call b,"},{"Start":"03:36.140 ","End":"03:40.580","Text":"so we\u0027ll take b towards infinity from 0."},{"Start":"03:40.580 ","End":"03:45.230","Text":"It\u0027s the same thing, minus 2x e^minus x squared"},{"Start":"03:45.230 ","End":"03:50.300","Text":"dx which is equal to the limit as b goes to infinity."},{"Start":"03:50.300 ","End":"03:53.510","Text":"Now we already figured out in the one asterisk that the"},{"Start":"03:53.510 ","End":"03:57.170","Text":"integral of this is e^minus x squared,"},{"Start":"03:57.170 ","End":"04:00.815","Text":"so we have e^minus x squared."},{"Start":"04:00.815 ","End":"04:04.685","Text":"This time with limits zero and b,"},{"Start":"04:04.685 ","End":"04:07.460","Text":"this equals just like we did here."},{"Start":"04:07.460 ","End":"04:09.490","Text":"If we put in 0, it\u0027s 1,"},{"Start":"04:09.490 ","End":"04:12.785","Text":"if we put in b, e to the minus b squared."},{"Start":"04:12.785 ","End":"04:16.655","Text":"So we have e to the minus b squared,"},{"Start":"04:16.655 ","End":"04:17.990","Text":"minus, like we said,"},{"Start":"04:17.990 ","End":"04:20.000","Text":"if we put in 0, we get 1,"},{"Start":"04:20.000 ","End":"04:24.155","Text":"and again the limit as b goes to infinity."},{"Start":"04:24.155 ","End":"04:27.245","Text":"This we already know comes out to 0,"},{"Start":"04:27.245 ","End":"04:31.580","Text":"0 minus 1 and now that gives us the double asterisk."},{"Start":"04:31.580 ","End":"04:34.265","Text":"If I go back up here,"},{"Start":"04:34.265 ","End":"04:37.375","Text":"this is equal to negative 1."},{"Start":"04:37.375 ","End":"04:44.240","Text":"Altogether, I get 1 plus negative 1, which is 0."},{"Start":"04:44.240 ","End":"04:47.935","Text":"Altogether, my integral comes out 0."},{"Start":"04:47.935 ","End":"04:52.220","Text":"Now all that remains is little bit of diagram for those who"},{"Start":"04:52.220 ","End":"04:57.020","Text":"studied the relation between definite integrals and area to make room for the graph,"},{"Start":"04:57.020 ","End":"04:58.700","Text":"I\u0027ve erased the other stuff."},{"Start":"04:58.700 ","End":"05:00.830","Text":"Now let\u0027s draw a pair of axes."},{"Start":"05:00.830 ","End":"05:08.985","Text":"Now, I want to draw the graph of the function y equals minus 2x e^minus x squared."},{"Start":"05:08.985 ","End":"05:11.225","Text":"Here\u0027s the rough shape of the graph."},{"Start":"05:11.225 ","End":"05:14.630","Text":"What we did in the one asterisk case where we went up to"},{"Start":"05:14.630 ","End":"05:19.610","Text":"0 is basically we computed this area here,"},{"Start":"05:19.610 ","End":"05:22.430","Text":"which I\u0027ve highlighted in blue."},{"Start":"05:22.430 ","End":"05:26.660","Text":"What we did here was not exactly to compute this area,"},{"Start":"05:26.660 ","End":"05:30.005","Text":"we actually computed negative the area."},{"Start":"05:30.005 ","End":"05:32.525","Text":"The area in geometry is always positive."},{"Start":"05:32.525 ","End":"05:34.025","Text":"When a function is positive,"},{"Start":"05:34.025 ","End":"05:36.785","Text":"the definite integral and the area match."},{"Start":"05:36.785 ","End":"05:39.155","Text":"But when the functions below the axis,"},{"Start":"05:39.155 ","End":"05:42.800","Text":"the integral is actually negative the area,"},{"Start":"05:42.800 ","End":"05:44.945","Text":"the integral comes out to be negative."},{"Start":"05:44.945 ","End":"05:50.255","Text":"So what we did when we got the integral was that we got that here,"},{"Start":"05:50.255 ","End":"05:52.835","Text":"the area was equal to 1,"},{"Start":"05:52.835 ","End":"05:57.290","Text":"and here we got that the integral was equal to minus 1,"},{"Start":"05:57.290 ","End":"06:03.095","Text":"but the integral in these cases is minus the area so the area here is also equal to 1."},{"Start":"06:03.095 ","End":"06:06.060","Text":"That\u0027s why when we added the 1 to the minus 1,"},{"Start":"06:06.060 ","End":"06:08.760","Text":"we got 0, we didn\u0027t get this area."},{"Start":"06:08.760 ","End":"06:13.280","Text":"In some way, the green canceled out the blue because it\u0027s like negative area."},{"Start":"06:13.280 ","End":"06:15.295","Text":"That explains the 0."},{"Start":"06:15.295 ","End":"06:17.300","Text":"Zero\u0027s made up of two opposing,"},{"Start":"06:17.300 ","End":"06:20.705","Text":"the same mass above the axis as below the axis."},{"Start":"06:20.705 ","End":"06:24.420","Text":"We\u0027re done with Type 1 improper integrals."}],"ID":9003},{"Watched":false,"Name":"Improper Integrals - Type 2","Duration":"4m 53s","ChapterTopicVideoID":8819,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"Continuing with improper integrals,"},{"Start":"00:02.655 ","End":"00:05.625","Text":"we\u0027ve done with type 1 and now let\u0027s get over the type 2."},{"Start":"00:05.625 ","End":"00:07.425","Text":"Type 2 is a bit more subtle,"},{"Start":"00:07.425 ","End":"00:12.780","Text":"at least it\u0027s not easy to identify initially because a type 2 improper integral looks"},{"Start":"00:12.780 ","End":"00:18.270","Text":"exactly like the integral from a to b of f of x dx."},{"Start":"00:18.270 ","End":"00:20.460","Text":"In type 1 we could recognize it because it was"},{"Start":"00:20.460 ","End":"00:23.700","Text":"an infinity here or minus infinity here or both,"},{"Start":"00:23.700 ","End":"00:25.920","Text":"but here it looks exactly the same."},{"Start":"00:25.920 ","End":"00:29.355","Text":"What actually makes it an improper integral?"},{"Start":"00:29.355 ","End":"00:35.415","Text":"Condition 1 is that f is undefined for some of the points in the interval."},{"Start":"00:35.415 ","End":"00:39.345","Text":"We\u0027re talking about the interval where a is less than or equal to x,"},{"Start":"00:39.345 ","End":"00:40.775","Text":"less than or equal to b,"},{"Start":"00:40.775 ","End":"00:47.190","Text":"so f is undefined at least one point in this interval a,"},{"Start":"00:47.190 ","End":"00:48.845","Text":"b, one or more."},{"Start":"00:48.845 ","End":"00:53.600","Text":"Not only is it undefined at these points it has to also meet further condition,"},{"Start":"00:53.600 ","End":"00:55.085","Text":"that at each of these points,"},{"Start":"00:55.085 ","End":"00:56.990","Text":"f has to be unbounded."},{"Start":"00:56.990 ","End":"00:59.470","Text":"Most students don\u0027t like the term unbounded,"},{"Start":"00:59.470 ","End":"01:01.940","Text":"so I\u0027ll give you a practical equivalent."},{"Start":"01:01.940 ","End":"01:04.850","Text":"Unbounded will be approximately equal to,"},{"Start":"01:04.850 ","End":"01:09.655","Text":"f of x tends to plus or minus infinity at these points."},{"Start":"01:09.655 ","End":"01:11.910","Text":"There are certain points, could be just one,"},{"Start":"01:11.910 ","End":"01:14.525","Text":"in a,b, could be at the end points, could be in the middle,"},{"Start":"01:14.525 ","End":"01:19.780","Text":"there is unbounded but in practice it means that these points when x goes to that point,"},{"Start":"01:19.780 ","End":"01:22.204","Text":"f goes to plus or minus infinity."},{"Start":"01:22.204 ","End":"01:23.689","Text":"Let\u0027s see an example."},{"Start":"01:23.689 ","End":"01:31.320","Text":"The integral from 1-10 of 1 over x minus 4 dx."},{"Start":"01:31.320 ","End":"01:36.994","Text":"Everything looks fine just like a normal integral until we noticed that at x equals 4,"},{"Start":"01:36.994 ","End":"01:38.975","Text":"the function is not defined,"},{"Start":"01:38.975 ","End":"01:40.175","Text":"and not only that,"},{"Start":"01:40.175 ","End":"01:45.605","Text":"when x goes to 4 the function goes to plus or minus infinity."},{"Start":"01:45.605 ","End":"01:52.110","Text":"We meet both these conditions because at x equals 4, f is undefined,"},{"Start":"01:52.110 ","End":"01:55.340","Text":"and furthermore f is unbounded because like I said,"},{"Start":"01:55.340 ","End":"01:57.575","Text":"if it goes to plus or minus infinity,"},{"Start":"01:57.575 ","End":"02:00.170","Text":"that counts as unbounded."},{"Start":"02:00.170 ","End":"02:03.725","Text":"This means that this is a type 2 improper integral."},{"Start":"02:03.725 ","End":"02:08.030","Text":"Another example, the integral of 1 over"},{"Start":"02:08.030 ","End":"02:14.450","Text":"sine x dx from x equals 0 to x equals 1."},{"Start":"02:14.450 ","End":"02:18.560","Text":"Here the problem is x equals 0,"},{"Start":"02:18.560 ","End":"02:20.420","Text":"because 0 is not defined,"},{"Start":"02:20.420 ","End":"02:22.865","Text":"sine 0 is 0, it\u0027s 1 over 0."},{"Start":"02:22.865 ","End":"02:25.474","Text":"But what\u0027s more when x equals 0,"},{"Start":"02:25.474 ","End":"02:32.180","Text":"not only is f undefined but since 1 over sine x goes to infinity,"},{"Start":"02:32.180 ","End":"02:35.825","Text":"then f is unbounded also."},{"Start":"02:35.825 ","End":"02:39.710","Text":"The last example is the integral from minus"},{"Start":"02:39.710 ","End":"02:46.650","Text":"1-0 of 1 over e^x minus 1 dx."},{"Start":"02:46.650 ","End":"02:50.600","Text":"Here again, the problem is x equals 0."},{"Start":"02:50.600 ","End":"02:54.680","Text":"What happens is that when x is 0 e^0 is 1."},{"Start":"02:54.680 ","End":"02:57.905","Text":"1 minus 1 gives us a 0 in the denominator,"},{"Start":"02:57.905 ","End":"03:05.885","Text":"so f is undefined and f is unbounded because when x goes to 0,"},{"Start":"03:05.885 ","End":"03:09.680","Text":"from below, e^x is slightly less than 1."},{"Start":"03:09.680 ","End":"03:10.880","Text":"This is slightly negative."},{"Start":"03:10.880 ","End":"03:12.910","Text":"It goes to minus infinity,"},{"Start":"03:12.910 ","End":"03:19.535","Text":"and since f goes to minus infinity f is also unbounded around the x equals 0."},{"Start":"03:19.535 ","End":"03:21.575","Text":"We\u0027ve seen three different cases,"},{"Start":"03:21.575 ","End":"03:24.440","Text":"they all had one problem point to where it\u0027s undefined."},{"Start":"03:24.440 ","End":"03:27.380","Text":"This one was in the middle of the interval domain,"},{"Start":"03:27.380 ","End":"03:29.840","Text":"this one was at the lower end point,"},{"Start":"03:29.840 ","End":"03:31.745","Text":"and this was at the upper end point."},{"Start":"03:31.745 ","End":"03:33.590","Text":"The all three could happen and we could also have"},{"Start":"03:33.590 ","End":"03:36.250","Text":"several of these points in the interval."},{"Start":"03:36.250 ","End":"03:41.420","Text":"But I\u0027d like you to contrast these three integrals with the following integral."},{"Start":"03:41.420 ","End":"03:48.740","Text":"The integral from 0-1 of sine x over x dx."},{"Start":"03:48.740 ","End":"03:51.640","Text":"There is a problem with x equals 0."},{"Start":"03:51.640 ","End":"03:55.295","Text":"In fact, the function is not defined at x equals 0,"},{"Start":"03:55.295 ","End":"04:00.380","Text":"so f is undefined at x equals 0."},{"Start":"04:00.380 ","End":"04:08.405","Text":"However, f is not unbounded because sine x over x has a limit of 1,"},{"Start":"04:08.405 ","End":"04:13.875","Text":"f of x tends to 1 when x goes to 0."},{"Start":"04:13.875 ","End":"04:16.295","Text":"When it has a limit it can\u0027t be unbounded,"},{"Start":"04:16.295 ","End":"04:18.220","Text":"it has a finite limit."},{"Start":"04:18.220 ","End":"04:21.905","Text":"This is not an improper type 2 integral."},{"Start":"04:21.905 ","End":"04:25.160","Text":"It\u0027s not exactly a regular integral either."},{"Start":"04:25.160 ","End":"04:28.370","Text":"But if you define f of 0,"},{"Start":"04:28.370 ","End":"04:32.000","Text":"just define it to be equal to 1."},{"Start":"04:32.000 ","End":"04:34.700","Text":"That\u0027s what we call removable discontinuity,"},{"Start":"04:34.700 ","End":"04:38.375","Text":"then the function will be continuous and then we can talk about it integrals."},{"Start":"04:38.375 ","End":"04:40.580","Text":"If we just define 0 to be 1,"},{"Start":"04:40.580 ","End":"04:43.745","Text":"then the integral is regular."},{"Start":"04:43.745 ","End":"04:45.560","Text":"There is no problem at 0 then."},{"Start":"04:45.560 ","End":"04:48.200","Text":"It\u0027ll be defined and won\u0027t be unbounded."},{"Start":"04:48.200 ","End":"04:53.850","Text":"Next, we\u0027ll see how to compute the integrals of type 2."}],"ID":9004},{"Watched":false,"Name":"Improper Integrals - Type 2 - Example 1","Duration":"3m 26s","ChapterTopicVideoID":8816,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"To learn how to solve improper integrals of type 2,"},{"Start":"00:04.050 ","End":"00:06.045","Text":"it\u0027s best to do some examples."},{"Start":"00:06.045 ","End":"00:08.265","Text":"Let\u0027s start with our first example,"},{"Start":"00:08.265 ","End":"00:15.585","Text":"which will be the integral from 0 to 1 of 1 over x dx."},{"Start":"00:15.585 ","End":"00:18.040","Text":"Now, why is this improper?"},{"Start":"00:18.040 ","End":"00:23.450","Text":"The answer is that in the range where 0 is less than or equal to x,"},{"Start":"00:23.450 ","End":"00:25.415","Text":"less than or equal to 1 or the domain,"},{"Start":"00:25.415 ","End":"00:27.590","Text":"there is a point, as you can see, 0,"},{"Start":"00:27.590 ","End":"00:32.405","Text":"where the function is not defined and it\u0027s also unbounded."},{"Start":"00:32.405 ","End":"00:34.615","Text":"At x equals 0,"},{"Start":"00:34.615 ","End":"00:38.140","Text":"we have the 2 conditions for the improper undefined,"},{"Start":"00:38.140 ","End":"00:40.250","Text":"I mean, we don\u0027t have 1 over 0."},{"Start":"00:40.250 ","End":"00:42.500","Text":"It\u0027s also unbounded,"},{"Start":"00:42.500 ","End":"00:46.220","Text":"which is like saying that the function goes to plus or minus infinity."},{"Start":"00:46.220 ","End":"00:51.260","Text":"Indeed it does because when x goes to 0 from above,"},{"Start":"00:51.260 ","End":"00:53.615","Text":"because we\u0027re only defined between 0 and 1."},{"Start":"00:53.615 ","End":"00:56.465","Text":"If x goes to 0 and slightly positive,"},{"Start":"00:56.465 ","End":"00:58.655","Text":"then 1 over x gets very large,"},{"Start":"00:58.655 ","End":"01:00.635","Text":"so it goes to infinity in fact."},{"Start":"01:00.635 ","End":"01:03.350","Text":"Now that we know that it\u0027s improper type 2,"},{"Start":"01:03.350 ","End":"01:05.180","Text":"how do we go about solving it?"},{"Start":"01:05.180 ","End":"01:07.790","Text":"Well very similar to the examples with infinity."},{"Start":"01:07.790 ","End":"01:09.020","Text":"We replace one of the limits,"},{"Start":"01:09.020 ","End":"01:12.380","Text":"the problematic one with a variable and that it tends to 0."},{"Start":"01:12.380 ","End":"01:14.425","Text":"What I\u0027m saying is we\u0027ll take the limit,"},{"Start":"01:14.425 ","End":"01:17.075","Text":"the lower limit instead of 0 we\u0027ll call it a,"},{"Start":"01:17.075 ","End":"01:19.070","Text":"and I\u0027ll let a go to 0."},{"Start":"01:19.070 ","End":"01:23.605","Text":"I have the integral from a to 1 of the same thing."},{"Start":"01:23.605 ","End":"01:26.990","Text":"Then I\u0027ll let a go to 0 but notice small point that"},{"Start":"01:26.990 ","End":"01:30.590","Text":"a doesn\u0027t just go to 0 because if I just say it goes to 0,"},{"Start":"01:30.590 ","End":"01:33.650","Text":"it could be from the positive side or the negative side,"},{"Start":"01:33.650 ","End":"01:35.825","Text":"but a can only be positive,"},{"Start":"01:35.825 ","End":"01:40.870","Text":"can be 0.000001, but it can\u0027t be minus that."},{"Start":"01:40.870 ","End":"01:43.460","Text":"In other words, what I\u0027m saying is we have to write 0 plus"},{"Start":"01:43.460 ","End":"01:45.710","Text":"because we can only approach 0 from the right."},{"Start":"01:45.710 ","End":"01:47.360","Text":"All we have to do is solve this,"},{"Start":"01:47.360 ","End":"01:49.490","Text":"first of all the integral and then we\u0027ll take the limit."},{"Start":"01:49.490 ","End":"01:55.175","Text":"This is the limit as a goes to 0 plus the integral of 1 over x."},{"Start":"01:55.175 ","End":"02:00.320","Text":"The anti-derivative or the indefinite integral is natural log of x."},{"Start":"02:00.320 ","End":"02:04.695","Text":"Then we take this between 1 and a."},{"Start":"02:04.695 ","End":"02:08.600","Text":"Now, that means that it\u0027s natural log of 1 minus natural log of a."},{"Start":"02:08.600 ","End":"02:12.320","Text":"Natural log of 1, as everyone knows, is 0."},{"Start":"02:12.320 ","End":"02:15.320","Text":"Natural log of a is just natural log of a."},{"Start":"02:15.320 ","End":"02:20.495","Text":"After take the limit as a goes to 0 from the right."},{"Start":"02:20.495 ","End":"02:27.050","Text":"This just equals minus the natural log of 0 plus,"},{"Start":"02:27.050 ","End":"02:31.370","Text":"and the natural log of 0 plus is known to be minus infinity."},{"Start":"02:31.370 ","End":"02:33.635","Text":"If this thing is minus infinity,"},{"Start":"02:33.635 ","End":"02:36.810","Text":"then the minus minus of it is infinity."},{"Start":"02:36.810 ","End":"02:40.070","Text":"In this case, we don\u0027t have a finite limit."},{"Start":"02:40.070 ","End":"02:44.870","Text":"We say that the integral diverges or diverges to infinity."},{"Start":"02:44.870 ","End":"02:46.580","Text":"If it was a finite number,"},{"Start":"02:46.580 ","End":"02:48.185","Text":"we would say it converges,"},{"Start":"02:48.185 ","End":"02:49.895","Text":"but that\u0027s not the case."},{"Start":"02:49.895 ","End":"02:52.955","Text":"Let\u0027s take a look at the geometric side of things."},{"Start":"02:52.955 ","End":"02:55.085","Text":"I\u0027ll draw some axis."},{"Start":"02:55.085 ","End":"02:59.450","Text":"Now, I\u0027ll try and sketch 1 over x, something like this."},{"Start":"02:59.450 ","End":"03:02.555","Text":"The 0.1, it would be here and 0\u0027s here."},{"Start":"03:02.555 ","End":"03:05.075","Text":"The integral from 0 to 1 would be,"},{"Start":"03:05.075 ","End":"03:06.470","Text":"if I sketch it,"},{"Start":"03:06.470 ","End":"03:12.500","Text":"it\u0027s this area here and all the way to infinity."},{"Start":"03:12.500 ","End":"03:17.510","Text":"It turns out that this area is infinite."},{"Start":"03:17.510 ","End":"03:20.270","Text":"Sometimes an area can be infinite in length,"},{"Start":"03:20.270 ","End":"03:21.800","Text":"but still be finite."},{"Start":"03:21.800 ","End":"03:23.120","Text":"We\u0027ve had that before,"},{"Start":"03:23.120 ","End":"03:24.320","Text":"but this time it\u0027s infinite,"},{"Start":"03:24.320 ","End":"03:26.760","Text":"there\u0027s nothing much more I can say."}],"ID":9005},{"Watched":false,"Name":"Improper Integrals - Type 2 - Example 2","Duration":"5m 43s","ChapterTopicVideoID":8817,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Now another example, this time,"},{"Start":"00:03.000 ","End":"00:07.980","Text":"let\u0027s take the integral from 1-4 of 1 over"},{"Start":"00:07.980 ","End":"00:13.770","Text":"the square root of 4 minus x, d_x."},{"Start":"00:13.770 ","End":"00:16.410","Text":"What\u0027s the problem here, if any?"},{"Start":"00:16.410 ","End":"00:17.940","Text":"Well, if you look at it,"},{"Start":"00:17.940 ","End":"00:25.050","Text":"it\u0027s not actually defined for all the points in the domain because when x is 4,"},{"Start":"00:25.050 ","End":"00:27.255","Text":"4 minus 4 is 0,"},{"Start":"00:27.255 ","End":"00:30.780","Text":"and we\u0027ve got a square root of 0 is 0 in the denominator,"},{"Start":"00:30.780 ","End":"00:32.895","Text":"so it\u0027s not defined there."},{"Start":"00:32.895 ","End":"00:36.450","Text":"It\u0027s like we said, there are 2 conditions to be a type 2 integral."},{"Start":"00:36.450 ","End":"00:39.570","Text":"Is that at least 1, or more points in the interval, in this case,"},{"Start":"00:39.570 ","End":"00:44.300","Text":"x equals 4, we have both undefined and unbounded."},{"Start":"00:44.300 ","End":"00:48.650","Text":"Unbounded, if x is very close to 4,"},{"Start":"00:48.650 ","End":"00:52.965","Text":"4 minus x is positive but very small,"},{"Start":"00:52.965 ","End":"00:54.315","Text":"in fact as small as we like."},{"Start":"00:54.315 ","End":"00:57.500","Text":"1 over something as small as we like will be as large as we like."},{"Start":"00:57.500 ","End":"01:01.970","Text":"In fact, it goes to infinity when x goes to 4 from below."},{"Start":"01:01.970 ","End":"01:04.470","Text":"Notice that because we\u0027re between 1 and 4,"},{"Start":"01:04.470 ","End":"01:06.910","Text":"x can only be less than 4."},{"Start":"01:06.910 ","End":"01:08.950","Text":"It can\u0027t be larger than 4."},{"Start":"01:08.950 ","End":"01:12.080","Text":"The way to solve this is very similar to the way we did when we"},{"Start":"01:12.080 ","End":"01:15.425","Text":"had the lower integration limit was bad."},{"Start":"01:15.425 ","End":"01:16.550","Text":"When the upper 1 was,"},{"Start":"01:16.550 ","End":"01:19.880","Text":"we just take the 4 away and put some letter b there."},{"Start":"01:19.880 ","End":"01:23.900","Text":"I take the integral from 1 to b of the same thing,"},{"Start":"01:23.900 ","End":"01:32.780","Text":"and then I take the limit as b goes to 4 of the same thing,"},{"Start":"01:32.780 ","End":"01:38.270","Text":"1 over the square root of 4 minus x, d_x."},{"Start":"01:38.270 ","End":"01:40.160","Text":"In order to continue,"},{"Start":"01:40.160 ","End":"01:44.840","Text":"I\u0027m going to need the indefinite integral or the antiderivative of"},{"Start":"01:44.840 ","End":"01:50.209","Text":"1 over the square root of 4 minus x. I\u0027ll do this as an exercise at the side."},{"Start":"01:50.209 ","End":"01:54.720","Text":"For here, I\u0027ll compute the indefinite integral of this function here,"},{"Start":"01:54.720 ","End":"02:04.520","Text":"and I\u0027ll rewrite it as the integral of 4 minus x to the power of minus a 1/2 d_x."},{"Start":"02:04.520 ","End":"02:13.860","Text":"There is a general rule which says that if I have a_x plus b to the power of n d_x,"},{"Start":"02:13.860 ","End":"02:17.795","Text":"and assuming that n is not equal to minus 1,"},{"Start":"02:17.795 ","End":"02:25.115","Text":"then this equals a_x plus b to the power of n plus 1."},{"Start":"02:25.115 ","End":"02:28.550","Text":"We divide it by n plus 1."},{"Start":"02:28.550 ","End":"02:33.450","Text":"But we also have to divide by 1 over a,"},{"Start":"02:33.580 ","End":"02:38.015","Text":"and that\u0027s what happens when we replace x by a_x plus b."},{"Start":"02:38.015 ","End":"02:40.340","Text":"We have to remember to divide by a."},{"Start":"02:40.340 ","End":"02:42.830","Text":"Of course, we add a C. But in"},{"Start":"02:42.830 ","End":"02:45.800","Text":"our case is not going to matter because when we use it for definite integral,"},{"Start":"02:45.800 ","End":"02:47.060","Text":"the C doesn\u0027t matter."},{"Start":"02:47.060 ","End":"02:52.955","Text":"Using this rule here, with n equals minus 1/2 and a equaling minus 1,"},{"Start":"02:52.955 ","End":"02:55.610","Text":"a is the coefficient of x, which is minus 1."},{"Start":"02:55.610 ","End":"02:59.705","Text":"We get 1 over minus 1 and"},{"Start":"02:59.705 ","End":"03:06.155","Text":"4 minus x to the power of n plus 1 is minus 1/2 plus 1,"},{"Start":"03:06.155 ","End":"03:10.570","Text":"so it\u0027s plus 1/2 over 1/2."},{"Start":"03:10.570 ","End":"03:15.650","Text":"Altogether, this equals if I take the minus 1/2 from the denominator,"},{"Start":"03:15.650 ","End":"03:17.345","Text":"becomes 2 in the numerator,"},{"Start":"03:17.345 ","End":"03:18.920","Text":"we get minus 2,"},{"Start":"03:18.920 ","End":"03:21.740","Text":"and then I can go back to the square root form,"},{"Start":"03:21.740 ","End":"03:26.280","Text":"square root of 4 minus x."},{"Start":"03:26.280 ","End":"03:29.629","Text":"Now I\u0027ve got the indefinite integral."},{"Start":"03:29.629 ","End":"03:35.585","Text":"Now I can come back here and say that this equals the limit"},{"Start":"03:35.585 ","End":"03:42.965","Text":"as b goes to 4 of minus 2 square root of 4 minus x."},{"Start":"03:42.965 ","End":"03:47.675","Text":"This has to be taken between the limits of 1 and b,"},{"Start":"03:47.675 ","End":"03:50.750","Text":"meaning substitute b, substitute 1, and subtract."},{"Start":"03:50.750 ","End":"03:56.190","Text":"What we get is the limit b goes to 4."},{"Start":"03:56.190 ","End":"03:58.730","Text":"Oh, there was some delicate point I forgot to mention,"},{"Start":"03:58.730 ","End":"04:00.154","Text":"and it\u0027s very important."},{"Start":"04:00.154 ","End":"04:04.145","Text":"When we go to 4, we have to stay inside the range,"},{"Start":"04:04.145 ","End":"04:07.415","Text":"and in the range, we\u0027re always less than or equal to 4."},{"Start":"04:07.415 ","End":"04:10.220","Text":"This forces us to take a 1 sided limit,"},{"Start":"04:10.220 ","End":"04:11.860","Text":"4 from the left."},{"Start":"04:11.860 ","End":"04:17.220","Text":"If I substitute b, I get minus 2 square root 4,"},{"Start":"04:17.220 ","End":"04:19.910","Text":"minus b, and when I put 1 in,"},{"Start":"04:19.910 ","End":"04:23.135","Text":"I get minus 2 square root of 3."},{"Start":"04:23.135 ","End":"04:26.900","Text":"That becomes plus 2 square root of 3, and now,"},{"Start":"04:26.900 ","End":"04:30.980","Text":"when b goes to 4, doesn\u0027t matter from which side then this thing,"},{"Start":"04:30.980 ","End":"04:32.210","Text":"4 minus 4 is 0."},{"Start":"04:32.210 ","End":"04:37.955","Text":"This thing disappears, and all we\u0027re left with is twice the square root of 3,"},{"Start":"04:37.955 ","End":"04:41.150","Text":"and that\u0027s our answer for this integral."},{"Start":"04:41.150 ","End":"04:44.300","Text":"What we can now say is that this integral converges,"},{"Start":"04:44.300 ","End":"04:45.620","Text":"or even more precisely,"},{"Start":"04:45.620 ","End":"04:48.995","Text":"this integral converges to twice the square root of 3."},{"Start":"04:48.995 ","End":"04:51.700","Text":"The only thing more is I\u0027d like to show you a picture for those who\u0027ve"},{"Start":"04:51.700 ","End":"04:55.110","Text":"studied the relation between definite integrals and area."},{"Start":"04:55.110 ","End":"04:57.785","Text":"I\u0027ll just draw a little sketch,"},{"Start":"04:57.785 ","End":"05:02.350","Text":"x equals 1, x equals 4,"},{"Start":"05:02.350 ","End":"05:05.540","Text":"the graph of 1 over the square root of 4 minus x."},{"Start":"05:05.540 ","End":"05:07.865","Text":"First of all, there\u0027s an asymptote here,"},{"Start":"05:07.865 ","End":"05:11.430","Text":"and it looks something like this,"},{"Start":"05:11.430 ","End":"05:13.110","Text":"and goes towards the asymptote,"},{"Start":"05:13.110 ","End":"05:15.675","Text":"and here it goes towards 0, but that doesn\u0027t matter."},{"Start":"05:15.675 ","End":"05:18.095","Text":"The points that matter are 1 and 4,"},{"Start":"05:18.095 ","End":"05:21.020","Text":"and what we basically have is the area"},{"Start":"05:21.020 ","End":"05:24.905","Text":"between the graph and the x-axis and between 1 and 4,"},{"Start":"05:24.905 ","End":"05:28.970","Text":"and although this thing goes up to infinity, still,"},{"Start":"05:28.970 ","End":"05:32.030","Text":"the area is finite, and in fact,"},{"Start":"05:32.030 ","End":"05:37.115","Text":"the area is equal to twice the square root of 3."},{"Start":"05:37.115 ","End":"05:41.570","Text":"That\u0027s the geometrical significance of this result."},{"Start":"05:41.570 ","End":"05:44.280","Text":"We\u0027re done with this example."}],"ID":9006},{"Watched":false,"Name":"Improper Integrals - Type 2 - Example 3","Duration":"4m 51s","ChapterTopicVideoID":8818,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"Now another example of an improper integral of type 2."},{"Start":"00:05.040 ","End":"00:13.185","Text":"This time our integral will be from minus 1 to 1 of 1 over x squared dx."},{"Start":"00:13.185 ","End":"00:16.450","Text":"As usual, it\u0027s not easy to spot right away,"},{"Start":"00:16.450 ","End":"00:18.320","Text":"but if we look at it,"},{"Start":"00:18.320 ","End":"00:19.940","Text":"and we look at our domain,"},{"Start":"00:19.940 ","End":"00:23.970","Text":"which is minus 1 less than or equal to x, less or equal to 1,"},{"Start":"00:23.970 ","End":"00:29.580","Text":"we easily see that at 0 we have a problem because at 0 the denominator is 0,"},{"Start":"00:29.580 ","End":"00:31.240","Text":"so it\u0027s not defined."},{"Start":"00:31.240 ","End":"00:34.470","Text":"At x equals 0 we have the 2 conditions fulfilled"},{"Start":"00:34.470 ","End":"00:37.050","Text":"for being a type 2 improper integral"},{"Start":"00:37.050 ","End":"00:41.385","Text":"and that is that the function is the 1 over x squared is undefined,"},{"Start":"00:41.385 ","End":"00:43.440","Text":"I\u0027m just writing this very briefly,"},{"Start":"00:43.440 ","End":"00:45.645","Text":"and it\u0027s also unbounded."},{"Start":"00:45.645 ","End":"00:49.100","Text":"Then I said the equivalent to unbounded is"},{"Start":"00:49.100 ","End":"00:54.200","Text":"if it tends to plus or minus infinity and indeed when x goes to 0,"},{"Start":"00:54.200 ","End":"00:58.640","Text":"it actually goes to plus infinity because the 0 is always positive."},{"Start":"00:58.640 ","End":"01:00.980","Text":"1 over positive 0 is plus infinity."},{"Start":"01:00.980 ","End":"01:04.315","Text":"Any event x equals 0 is a problem point,"},{"Start":"01:04.315 ","End":"01:06.890","Text":"in contrast to the 2 previous exercises"},{"Start":"01:06.890 ","End":"01:10.100","Text":"where we had a problem point at the top and at the bottom,"},{"Start":"01:10.100 ","End":"01:11.800","Text":"now we have a problem point in the middle."},{"Start":"01:11.800 ","End":"01:15.995","Text":"And when it\u0027s in the middle between the ends of the domain,"},{"Start":"01:15.995 ","End":"01:17.550","Text":"what we do is the following,"},{"Start":"01:17.550 ","End":"01:19.790","Text":"is we break it up into 2 integrals,"},{"Start":"01:19.790 ","End":"01:21.620","Text":"each with a problem at one end."},{"Start":"01:21.620 ","End":"01:24.605","Text":"Basically what we do is instead of minus 1 all the way to 1,"},{"Start":"01:24.605 ","End":"01:26.915","Text":"we go from minus 1 to 0,"},{"Start":"01:26.915 ","End":"01:30.920","Text":"and then we go from 0 to 1 of the same thing."},{"Start":"01:30.920 ","End":"01:39.275","Text":"1 over x squared dx plus the integral from 0 to 1 also of 1 over x squared dx."},{"Start":"01:39.275 ","End":"01:42.304","Text":"Each of these, we can use our usual techniques"},{"Start":"01:42.304 ","End":"01:45.650","Text":"like here putting instead of 0 a b which tends to 0,"},{"Start":"01:45.650 ","End":"01:48.590","Text":"and here putting an a which tends to 0,"},{"Start":"01:48.590 ","End":"01:51.650","Text":"but first, I\u0027d like to do a little side exercise"},{"Start":"01:51.650 ","End":"01:55.895","Text":"just to find the indefinite integral of 1 over x squared."},{"Start":"01:55.895 ","End":"02:00.230","Text":"The indefinite integral of 1 over x squared dx"},{"Start":"02:00.230 ","End":"02:07.745","Text":"is just the integral of x to the minus 2dx and using our rules for exponents,"},{"Start":"02:07.745 ","End":"02:12.740","Text":"this is equal to x to the minus 2 plus 1 is minus 1"},{"Start":"02:12.740 ","End":"02:16.190","Text":"and then we divide by this minus 1 and we add c."},{"Start":"02:16.190 ","End":"02:22.130","Text":"Altogether what we get is that this thing is equal to minus 1 over x."},{"Start":"02:22.130 ","End":"02:23.990","Text":"I\u0027m going to forget about the c,"},{"Start":"02:23.990 ","End":"02:26.810","Text":"because when we do definite integrals the c doesn\u0027t matter."},{"Start":"02:26.810 ","End":"02:28.220","Text":"Let\u0027s do them separately,"},{"Start":"02:28.220 ","End":"02:35.055","Text":"let\u0027s call this one asterisk and the second integral we\u0027ll call double asterisk,"},{"Start":"02:35.055 ","End":"02:38.770","Text":"and then after we\u0027ve computed each one we\u0027ll add them and we\u0027ll get the result."},{"Start":"02:38.770 ","End":"02:41.255","Text":"First of all, the asterisk,"},{"Start":"02:41.255 ","End":"02:46.629","Text":"so we go from minus 1, instead of 0, we go to b,"},{"Start":"02:46.629 ","End":"02:49.070","Text":"but then we take the limit,"},{"Start":"02:49.070 ","End":"02:53.095","Text":"the integral from minus 1 to b then b goes to 0."},{"Start":"02:53.095 ","End":"02:57.010","Text":"We do have the indefinite integral minus 1 over x,"},{"Start":"02:57.010 ","End":"03:02.825","Text":"so this is the limit as b goes to 0 of minus 1 over x,"},{"Start":"03:02.825 ","End":"03:07.430","Text":"taken between the limits of b and minus 1."},{"Start":"03:07.430 ","End":"03:09.515","Text":"Now if I put in b,"},{"Start":"03:09.515 ","End":"03:11.870","Text":"I get minus 1 over b."},{"Start":"03:11.870 ","End":"03:13.745","Text":"Minus 1 over b,"},{"Start":"03:13.745 ","End":"03:15.485","Text":"if I put in minus 1,"},{"Start":"03:15.485 ","End":"03:22.414","Text":"I get minus plus 1 and we need the limit of this as b goes to 0."},{"Start":"03:22.414 ","End":"03:24.830","Text":"Of course, I should have mentioned that"},{"Start":"03:24.830 ","End":"03:29.585","Text":"when we are doing this integral and we are between minus 1 and 0,"},{"Start":"03:29.585 ","End":"03:35.465","Text":"then b goes to 0 from the left because I\u0027m coming from the direction of minus 1."},{"Start":"03:35.465 ","End":"03:37.790","Text":"When I get the final answer,"},{"Start":"03:37.790 ","End":"03:39.785","Text":"if b is negative 0,"},{"Start":"03:39.785 ","End":"03:45.665","Text":"then this comes out minus 1 over negative 0 and so it\u0027s plus infinity."},{"Start":"03:45.665 ","End":"03:48.215","Text":"However, this is not a finite number,"},{"Start":"03:48.215 ","End":"03:49.775","Text":"and so this part,"},{"Start":"03:49.775 ","End":"03:51.980","Text":"the asterisk, does not converge,"},{"Start":"03:51.980 ","End":"03:54.065","Text":"it diverges to infinity."},{"Start":"03:54.065 ","End":"03:56.000","Text":"If 1 part of this divergences,"},{"Start":"03:56.000 ","End":"03:57.260","Text":"we don\u0027t even continue,"},{"Start":"03:57.260 ","End":"04:07.855","Text":"we immediately say that the integral from minus 1 to 1 of 1 over x squared dx diverges."},{"Start":"04:07.855 ","End":"04:10.800","Text":"That\u0027s it unless we want to draw a little sketch,"},{"Start":"04:10.800 ","End":"04:16.370","Text":"basically we have some axis and this function is always positive."},{"Start":"04:16.370 ","End":"04:19.070","Text":"This would be the y-axis, the x-axis,"},{"Start":"04:19.070 ","End":"04:21.740","Text":"1 over x squared looks something like this"},{"Start":"04:21.740 ","End":"04:25.430","Text":"with an asymptote here and over here it\u0027s just symmetrical."},{"Start":"04:25.430 ","End":"04:27.230","Text":"It goes up to infinity."},{"Start":"04:27.230 ","End":"04:28.130","Text":"This is an asymptote."},{"Start":"04:28.130 ","End":"04:29.135","Text":"This is an asymptote."},{"Start":"04:29.135 ","End":"04:34.410","Text":"What we had to do is figure out from minus 1 to plus 1"},{"Start":"04:34.410 ","End":"04:38.470","Text":"and so it turned out that we just tried to compute 1 part of it,"},{"Start":"04:38.470 ","End":"04:42.920","Text":"this part of it and already we got infinity here for the area"},{"Start":"04:42.920 ","End":"04:45.380","Text":"and it also turns out to be infinity here."},{"Start":"04:45.380 ","End":"04:47.890","Text":"If we did, the other one it wouldn\u0027t have made a difference."},{"Start":"04:47.890 ","End":"04:52.000","Text":"That\u0027s it we\u0027re done with type 2 proper integrals."}],"ID":9007},{"Watched":false,"Name":"Improper Integrals - Type 3","Duration":"2m 36s","ChapterTopicVideoID":8808,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"Now we come to the last type,"},{"Start":"00:02.190 ","End":"00:05.160","Text":"which is type 3 improper integral."},{"Start":"00:05.160 ","End":"00:06.900","Text":"It\u0027s not universally recognized."},{"Start":"00:06.900 ","End":"00:10.935","Text":"Some professors and books use the term, some don\u0027t."},{"Start":"00:10.935 ","End":"00:14.145","Text":"But in any case, this will appear in exams."},{"Start":"00:14.145 ","End":"00:16.200","Text":"Whether you call it this or not,"},{"Start":"00:16.200 ","End":"00:17.820","Text":"you should know about it."},{"Start":"00:17.820 ","End":"00:25.665","Text":"What it is is a combo which mixes type 1 and type 2 all in the same integral."},{"Start":"00:25.665 ","End":"00:27.945","Text":"Easiest, to show you an example,"},{"Start":"00:27.945 ","End":"00:36.870","Text":"an example would be the integral from 0 to infinity of 1 over x squared dx."},{"Start":"00:36.870 ","End":"00:38.420","Text":"Now, on the one hand,"},{"Start":"00:38.420 ","End":"00:40.400","Text":"because of this infinity,"},{"Start":"00:40.400 ","End":"00:44.255","Text":"that makes it into a type 1 improper integral."},{"Start":"00:44.255 ","End":"00:46.880","Text":"Because of the 0 here,"},{"Start":"00:46.880 ","End":"00:52.655","Text":"the 0 is a point in the domain where the function is not defined."},{"Start":"00:52.655 ","End":"00:54.590","Text":"Not only is it not defined,"},{"Start":"00:54.590 ","End":"00:56.120","Text":"but it\u0027s not bounded."},{"Start":"00:56.120 ","End":"01:00.245","Text":"In other words, x equals 0 is where the function is undefined."},{"Start":"01:00.245 ","End":"01:02.540","Text":"Those were the 2 conditions for type 2"},{"Start":"01:02.540 ","End":"01:06.859","Text":"undefined and it\u0027s unbounded because when x goes to 0,"},{"Start":"01:06.859 ","End":"01:09.980","Text":"then the function goes to infinity."},{"Start":"01:09.980 ","End":"01:11.795","Text":"If x goes to 0 from above,"},{"Start":"01:11.795 ","End":"01:14.945","Text":"1 over x squared is 1 over 0, which is infinity."},{"Start":"01:14.945 ","End":"01:19.100","Text":"This is a type 3, and the usual method is to just"},{"Start":"01:19.100 ","End":"01:23.300","Text":"split up the range of the integral so we only have 1 problem at a time."},{"Start":"01:23.300 ","End":"01:24.710","Text":"For example, in this case,"},{"Start":"01:24.710 ","End":"01:26.855","Text":"the integral from 0 to infinity,"},{"Start":"01:26.855 ","End":"01:31.490","Text":"we would break it up into the integral from 0 to say 1,"},{"Start":"01:31.490 ","End":"01:34.785","Text":"and the integral from 1 to infinity."},{"Start":"01:34.785 ","End":"01:38.510","Text":"Then we\u0027d have here 1 over x squared dx,"},{"Start":"01:38.510 ","End":"01:40.850","Text":"1 over x squared dx,"},{"Start":"01:40.850 ","End":"01:42.845","Text":"1 over x squared dx."},{"Start":"01:42.845 ","End":"01:49.200","Text":"That would make this a type 1 and this would become type 2."},{"Start":"01:49.200 ","End":"01:53.720","Text":"As usual, they both have to converge in order for us to say that this converges."},{"Start":"01:53.720 ","End":"01:55.865","Text":"If either 1 of these diverges,"},{"Start":"01:55.865 ","End":"01:58.160","Text":"then the integral is divergent."},{"Start":"01:58.160 ","End":"02:03.425","Text":"We\u0027ve encountered these before that this actually converges to some number."},{"Start":"02:03.425 ","End":"02:07.710","Text":"This one diverges specifically to infinity."},{"Start":"02:07.710 ","End":"02:10.530","Text":"Finite number plus infinity is infinity,"},{"Start":"02:10.530 ","End":"02:15.320","Text":"so we said that this also diverges to infinity as a matter of fact."},{"Start":"02:15.320 ","End":"02:17.390","Text":"That\u0027s basically all there is."},{"Start":"02:17.390 ","End":"02:21.520","Text":"Type 3 just has to be split up into smaller problems and"},{"Start":"02:21.520 ","End":"02:26.690","Text":"usual techniques apply here we take the integral from 1 to b and let b go to infinity."},{"Start":"02:26.690 ","End":"02:31.160","Text":"Here we take the integral from a to 1 and let a go to 0 and so on."},{"Start":"02:31.160 ","End":"02:35.095","Text":"But you should recognize that there is a type which is basically a combo type."},{"Start":"02:35.095 ","End":"02:37.450","Text":"Okay. That\u0027s all."}],"ID":9008},{"Watched":false,"Name":"P Integral","Duration":"1m 2s","ChapterTopicVideoID":8809,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.315","Text":"In this clip, I\u0027d like to introduce you to the P integral,"},{"Start":"00:03.315 ","End":"00:06.240","Text":"which is an improper integral of type 1."},{"Start":"00:06.240 ","End":"00:10.500","Text":"It looks like the integral from 1 to"},{"Start":"00:10.500 ","End":"00:17.145","Text":"infinity of 1 over x to the power of p dx."},{"Start":"00:17.145 ","End":"00:19.365","Text":"The p, of course, is what gives it its name."},{"Start":"00:19.365 ","End":"00:23.685","Text":"P is any finite number, positive or negative."},{"Start":"00:23.685 ","End":"00:26.670","Text":"What I\u0027d like you to know about this integral is"},{"Start":"00:26.670 ","End":"00:32.550","Text":"that there are 2 cases that this integral converges"},{"Start":"00:32.550 ","End":"00:38.505","Text":"if p is bigger than 1 and"},{"Start":"00:38.505 ","End":"00:44.585","Text":"diverges if p is less than or equal to 1."},{"Start":"00:44.585 ","End":"00:47.140","Text":"We\u0027re going to be using this in the following exercises,"},{"Start":"00:47.140 ","End":"00:49.340","Text":"so I thought I\u0027d better introduce it."},{"Start":"00:49.340 ","End":"00:55.050","Text":"When it converges, it converges to 1 over p minus 1,"},{"Start":"00:55.050 ","End":"00:56.540","Text":"and when it diverges,"},{"Start":"00:56.540 ","End":"01:02.280","Text":"it diverges to plus infinity. That\u0027s it."}],"ID":9009},{"Watched":false,"Name":"Exercise 1","Duration":"2m 30s","ChapterTopicVideoID":4562,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this exercise, we have to compute the"},{"Start":"00:02.940 ","End":"00:07.350","Text":"following integral if possible or say that it diverges."},{"Start":"00:07.350 ","End":"00:10.890","Text":"What I\u0027m going to need here is the"},{"Start":"00:10.890 ","End":"00:14.290","Text":"indefinite integral also before I do the definite integral."},{"Start":"00:14.290 ","End":"00:17.130","Text":"Let me do that as a computation at the side."},{"Start":"00:17.130 ","End":"00:19.035","Text":"I want, first of all,"},{"Start":"00:19.035 ","End":"00:24.040","Text":"just the indefinite integral of 1 over x^4 dx."},{"Start":"00:26.570 ","End":"00:33.480","Text":"This equals the integral of x^ minus 4 dx,"},{"Start":"00:33.480 ","End":"00:36.510","Text":"and this equals, according to the rules of exponents,"},{"Start":"00:36.510 ","End":"00:40.820","Text":"x^ minus 3 over minus 3."},{"Start":"00:40.820 ","End":"00:45.215","Text":"I\u0027m not going to put in the plus C because we\u0027re going to do a definite integral."},{"Start":"00:45.215 ","End":"00:52.714","Text":"I can just maybe rewrite it as minus 1 over 3x cubed."},{"Start":"00:52.714 ","End":"00:54.255","Text":"That\u0027s more convenient."},{"Start":"00:54.255 ","End":"00:56.130","Text":"For those who really insist on it,"},{"Start":"00:56.130 ","End":"00:57.570","Text":"I can put a plus C here,"},{"Start":"00:57.570 ","End":"00:59.590","Text":"but I don\u0027t need it."},{"Start":"00:59.720 ","End":"01:03.065","Text":"Now getting back here,"},{"Start":"01:03.065 ","End":"01:05.540","Text":"the way we do an improper integral,"},{"Start":"01:05.540 ","End":"01:08.720","Text":"and it\u0027s improper because of this infinity here,"},{"Start":"01:08.720 ","End":"01:12.065","Text":"is we replace the infinity by some number."},{"Start":"01:12.065 ","End":"01:15.310","Text":"I\u0027ll call it b since we usually take integrals from a to b."},{"Start":"01:15.310 ","End":"01:17.755","Text":"But here, I\u0027ll leave the 1 as is,"},{"Start":"01:17.755 ","End":"01:24.785","Text":"and what we do is we take the limit as b goes to infinity of the same thing,"},{"Start":"01:24.785 ","End":"01:28.795","Text":"1 over x^4 dx."},{"Start":"01:28.795 ","End":"01:34.640","Text":"Now I can use the result from here and say that this is equal to"},{"Start":"01:34.640 ","End":"01:43.410","Text":"minus 1 over 3x cubed taken between 1 and b."},{"Start":"01:45.350 ","End":"01:48.930","Text":"Substitute b, substitute 1, and subtract."},{"Start":"01:48.930 ","End":"01:58.155","Text":"Substitute b, I get minus 1 over 3b cubed minus."},{"Start":"01:58.155 ","End":"02:02.130","Text":"If you want to get minus 1 over 3,"},{"Start":"02:02.130 ","End":"02:03.660","Text":"1 cubed is just 1,"},{"Start":"02:03.660 ","End":"02:05.365","Text":"but okay, I\u0027ll leave it in."},{"Start":"02:05.365 ","End":"02:08.330","Text":"Sorry, I forgot to write the limit."},{"Start":"02:08.330 ","End":"02:11.450","Text":"Limit as b goes to infinity,"},{"Start":"02:11.450 ","End":"02:13.580","Text":"I better put some brackets here,"},{"Start":"02:13.580 ","End":"02:15.335","Text":"and now let\u0027s see."},{"Start":"02:15.335 ","End":"02:16.910","Text":"When b goes to infinity,"},{"Start":"02:16.910 ","End":"02:19.540","Text":"this clearly goes to zero."},{"Start":"02:19.540 ","End":"02:21.430","Text":"Here, we get zero."},{"Start":"02:21.430 ","End":"02:23.180","Text":"This is just a constant minus,"},{"Start":"02:23.180 ","End":"02:25.280","Text":"minus 1 over 3,"},{"Start":"02:25.280 ","End":"02:31.590","Text":"so the answer is simply 1/3. That\u0027s it."}],"ID":4571},{"Watched":false,"Name":"Exercise 2","Duration":"2m 6s","ChapterTopicVideoID":4563,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.900","Text":"In this exercise, we have to compute this improper integral provided it converges."},{"Start":"00:06.900 ","End":"00:12.375","Text":"What we do with improper integrals is we replace the infinity by a limit."},{"Start":"00:12.375 ","End":"00:15.435","Text":"What I want is the limit as,"},{"Start":"00:15.435 ","End":"00:17.610","Text":"let\u0027s use a letter, say b,"},{"Start":"00:17.610 ","End":"00:22.410","Text":"b is often used for the upper limit because we take it from a to b but it\u0027s"},{"Start":"00:22.410 ","End":"00:27.780","Text":"from 1-b of 1 over the square root of x dx."},{"Start":"00:27.780 ","End":"00:31.985","Text":"Before I continue, I want the indefinite integral, the anti-derivative."},{"Start":"00:31.985 ","End":"00:36.860","Text":"I\u0027m going to do that at the side and say that the integral of"},{"Start":"00:36.860 ","End":"00:41.660","Text":"1 over the square root of x dx is equal to."},{"Start":"00:41.660 ","End":"00:46.805","Text":"Now, I could use it with exponents and say x to the minus 1/2 but it\u0027s easiest for me."},{"Start":"00:46.805 ","End":"00:48.605","Text":"It looks so familiar."},{"Start":"00:48.605 ","End":"00:52.910","Text":"It\u0027s the derivative of square root of x,"},{"Start":"00:52.910 ","End":"00:54.500","Text":"so if I put a 2 here and here,"},{"Start":"00:54.500 ","End":"00:56.045","Text":"it should be okay."},{"Start":"00:56.045 ","End":"01:00.960","Text":"This is going to equal twice the square root of x plus a constant,"},{"Start":"01:00.960 ","End":"01:02.945","Text":"so that\u0027s the indefinite integral."},{"Start":"01:02.945 ","End":"01:06.515","Text":"Of course, with definite integral I needn\u0027t bother with the constant."},{"Start":"01:06.515 ","End":"01:13.215","Text":"I\u0027m just going to say that this is the limit as b goes to infinity."},{"Start":"01:13.215 ","End":"01:17.900","Text":"But this integral we\u0027ve already computed is twice the square root of"},{"Start":"01:17.900 ","End":"01:23.060","Text":"x but I need to take it between the bounds of 1 and b,"},{"Start":"01:23.060 ","End":"01:26.180","Text":"which means substitute this, substitute that and subtract."},{"Start":"01:26.180 ","End":"01:36.890","Text":"We get the limit of twice square root of b minus twice square root of 1,"},{"Start":"01:36.890 ","End":"01:39.470","Text":"b goes to infinity."},{"Start":"01:39.470 ","End":"01:44.180","Text":"Now, this thing is a constant just 2,"},{"Start":"01:44.180 ","End":"01:47.570","Text":"so what I get basically is twice the square root of"},{"Start":"01:47.570 ","End":"01:51.650","Text":"infinity minus 2 and the square root of infinity is infinity,"},{"Start":"01:51.650 ","End":"01:59.340","Text":"so the whole thing is infinity by which it means that this integral diverges."},{"Start":"01:59.340 ","End":"02:02.190","Text":"We have determined that it diverges,"},{"Start":"02:02.190 ","End":"02:06.900","Text":"so there is no answer for what is the value. Done."}],"ID":4572},{"Watched":false,"Name":"Exercise 3","Duration":"4m 16s","ChapterTopicVideoID":4564,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"Here we have an improper integral,"},{"Start":"00:01.860 ","End":"00:04.620","Text":"so we have to decide if it converges or diverges,"},{"Start":"00:04.620 ","End":"00:06.690","Text":"and if it converges, we have to compute it."},{"Start":"00:06.690 ","End":"00:08.295","Text":"I\u0027ve written it over here."},{"Start":"00:08.295 ","End":"00:11.055","Text":"What we do with an improper integral like this,"},{"Start":"00:11.055 ","End":"00:19.410","Text":"where the lower limit is minus infinity is we replace that lower limit by a,"},{"Start":"00:19.410 ","End":"00:23.340","Text":"and then we take the limit as a goes to minus infinity, well,"},{"Start":"00:23.340 ","End":"00:26.790","Text":"the limit of this when a goes to minus infinity is what it means to"},{"Start":"00:26.790 ","End":"00:30.960","Text":"take the lower limit minus infinity of the same integrant,"},{"Start":"00:30.960 ","End":"00:37.200","Text":"which is 1 over 2x minus 5 to the 5th dx."},{"Start":"00:37.200 ","End":"00:41.900","Text":"Before I continue, I\u0027d like to know what the indefinite integral,"},{"Start":"00:41.900 ","End":"00:43.430","Text":"the anti-derivative of this is,"},{"Start":"00:43.430 ","End":"00:45.410","Text":"and I\u0027ll do that at the side."},{"Start":"00:45.410 ","End":"00:47.030","Text":"What I want to know is,"},{"Start":"00:47.030 ","End":"00:56.730","Text":"what is the integral of 1 over 2x minus 5 to the 5th dx."},{"Start":"00:56.730 ","End":"00:59.750","Text":"This is going to equal the integral."},{"Start":"00:59.750 ","End":"01:08.245","Text":"I can write it as 2x minus 5 to the power of minus 5 dx."},{"Start":"01:08.245 ","End":"01:10.850","Text":"It\u0027s like x to the minus 5,"},{"Start":"01:10.850 ","End":"01:13.295","Text":"but instead of x, I have a linear expression."},{"Start":"01:13.295 ","End":"01:15.425","Text":"What I do is at first,"},{"Start":"01:15.425 ","End":"01:18.740","Text":"just integrate as if they were just x here."},{"Start":"01:18.740 ","End":"01:22.880","Text":"I get 2x minus 5,"},{"Start":"01:22.880 ","End":"01:24.380","Text":"I raise the power by 1,"},{"Start":"01:24.380 ","End":"01:29.015","Text":"so it\u0027s minus 4 and divide by minus 4."},{"Start":"01:29.015 ","End":"01:31.460","Text":"But because it\u0027s not x,"},{"Start":"01:31.460 ","End":"01:34.325","Text":"it\u0027s a linear expression, ax plus b."},{"Start":"01:34.325 ","End":"01:39.020","Text":"We divide by the a by the coefficient of x. I also have to divide by 2."},{"Start":"01:39.020 ","End":"01:41.225","Text":"I just put it on the bottom here."},{"Start":"01:41.225 ","End":"01:42.920","Text":"That\u0027s this 2 here."},{"Start":"01:42.920 ","End":"01:46.820","Text":"Perhaps I\u0027ll rewrite this a little more conveniently."},{"Start":"01:46.820 ","End":"01:51.095","Text":"This is 1 over 8,"},{"Start":"01:51.095 ","End":"01:53.675","Text":"4 times 2 is 8 and it\u0027s a minus,"},{"Start":"01:53.675 ","End":"01:57.370","Text":"minus 1/8 and 1 over"},{"Start":"01:57.370 ","End":"02:05.700","Text":"2x minus 5 to the power of plus 4 plus c. Now I\u0027ll get back over here."},{"Start":"02:05.700 ","End":"02:10.275","Text":"This is going to equal minus 1/8,"},{"Start":"02:10.275 ","End":"02:13.460","Text":"I can take that right in front of the limit."},{"Start":"02:13.460 ","End":"02:19.590","Text":"The limit as a goes to minus infinity."},{"Start":"02:19.590 ","End":"02:23.800","Text":"What I have to do is this is a definite integral, this is the indefinite,"},{"Start":"02:23.800 ","End":"02:29.720","Text":"so I just have to take from here 1 over 2x minus"},{"Start":"02:29.720 ","End":"02:36.560","Text":"5 to the 4th and evaluated between a and 0."},{"Start":"02:36.560 ","End":"02:37.790","Text":"I mean substitute 0,"},{"Start":"02:37.790 ","End":"02:40.780","Text":"substitute a and subtract."},{"Start":"02:40.780 ","End":"02:45.285","Text":"This is equal to minus 1/8 limit,"},{"Start":"02:45.285 ","End":"02:47.810","Text":"a goes to minus infinity."},{"Start":"02:47.810 ","End":"02:50.255","Text":"If I substitute 0,"},{"Start":"02:50.255 ","End":"02:57.150","Text":"I just get 1 over minus 5 to the 4th,"},{"Start":"02:57.150 ","End":"03:03.160","Text":"which is 625, so that\u0027s 1 over 625,"},{"Start":"03:03.160 ","End":"03:13.620","Text":"minus what happens if I put a here which is 1 over 2a minus 5 to the 4th."},{"Start":"03:13.620 ","End":"03:17.495","Text":"If I let a go to infinity, basically,"},{"Start":"03:17.495 ","End":"03:21.710","Text":"you can see that the denominator goes to infinity."},{"Start":"03:21.710 ","End":"03:24.320","Text":"Because if there goes to infinity,"},{"Start":"03:24.320 ","End":"03:27.665","Text":"then certainly to 2a minus 5 goes to infinity."},{"Start":"03:27.665 ","End":"03:30.845","Text":"If I raise it to the power of 4, it still infinity."},{"Start":"03:30.845 ","End":"03:34.975","Text":"This basically becomes minus 1/8,"},{"Start":"03:34.975 ","End":"03:44.745","Text":"and what I get is 1 over 625 minus 1 over infinity to the 4th."},{"Start":"03:44.745 ","End":"03:49.850","Text":"This thing is 0 because infinity to the 4th and infinity,"},{"Start":"03:49.850 ","End":"03:51.560","Text":"and 1 over infinity is 0,"},{"Start":"03:51.560 ","End":"03:54.595","Text":"so it\u0027s just 1/625."},{"Start":"03:54.595 ","End":"04:02.000","Text":"This is equal to minus 1/625 times 8."},{"Start":"04:02.000 ","End":"04:03.470","Text":"Let\u0027s see if I can compute that."},{"Start":"04:03.470 ","End":"04:05.690","Text":"If I double it, it\u0027s 1,250,"},{"Start":"04:05.690 ","End":"04:10.305","Text":"double it again, 2,500 double it again, 5,000."},{"Start":"04:10.305 ","End":"04:16.750","Text":"I make it minus 1 over 5,000. We\u0027re done."}],"ID":4573},{"Watched":false,"Name":"Exercise 4","Duration":"4m ","ChapterTopicVideoID":4565,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"We\u0027re given an integral, let me have to check if it converges and if it does converge,"},{"Start":"00:04.470 ","End":"00:06.360","Text":"we have to determine its value."},{"Start":"00:06.360 ","End":"00:08.355","Text":"I copied it over here."},{"Start":"00:08.355 ","End":"00:11.640","Text":"If this is an improper integral because of the infinity."},{"Start":"00:11.640 ","End":"00:14.370","Text":"The standard thing to do in this case,"},{"Start":"00:14.370 ","End":"00:18.060","Text":"is to take it as the limit when something goes to infinity."},{"Start":"00:18.060 ","End":"00:21.510","Text":"I\u0027ll call that something b. I\u0027ll take b goes to"},{"Start":"00:21.510 ","End":"00:25.080","Text":"infinity of this integral with the same lower limit"},{"Start":"00:25.080 ","End":"00:33.735","Text":"and the same integrant 1 over the cube root of 2x plus 1 dx."},{"Start":"00:33.735 ","End":"00:37.340","Text":"I\u0027m going to need the indefinite integral,"},{"Start":"00:37.340 ","End":"00:39.930","Text":"the anti-derivative before I continue."},{"Start":"00:39.930 ","End":"00:41.905","Text":"I\u0027ll do that at the side."},{"Start":"00:41.905 ","End":"00:46.640","Text":"What I want is the integral indefinite."},{"Start":"00:46.640 ","End":"00:50.975","Text":"I don\u0027t write limit of 1 over the cube root."},{"Start":"00:50.975 ","End":"00:53.510","Text":"Let\u0027s write it in terms of exponents."},{"Start":"00:53.510 ","End":"00:56.420","Text":"The cube root is to the power of a 1/3,"},{"Start":"00:56.420 ","End":"00:58.100","Text":"and it\u0027s on the denominator,"},{"Start":"00:58.100 ","End":"00:59.510","Text":"so it\u0027s minus a 1/3."},{"Start":"00:59.510 ","End":"01:06.680","Text":"What I have is 2x plus 1 to the power of minus a 1/3 dx."},{"Start":"01:06.680 ","End":"01:10.380","Text":"Here I use the rules for exponents."},{"Start":"01:10.970 ","End":"01:14.330","Text":"Forget that it\u0027s 2x plus 1 for a moment,"},{"Start":"01:14.330 ","End":"01:16.265","Text":"just like think of it like x."},{"Start":"01:16.265 ","End":"01:20.330","Text":"We raise the power by 1 and we get 2x plus"},{"Start":"01:20.330 ","End":"01:26.255","Text":"1 to the power of 2/3 because minus a 1/3 plus 1 is 2/3,"},{"Start":"01:26.255 ","End":"01:30.020","Text":"then we divide it by 2/3."},{"Start":"01:30.020 ","End":"01:33.920","Text":"But because it isn\u0027t x it\u0027s 2x plus 1,"},{"Start":"01:33.920 ","End":"01:35.659","Text":"which is the linear function."},{"Start":"01:35.659 ","End":"01:38.450","Text":"We can divide by the internal derivative."},{"Start":"01:38.450 ","End":"01:40.175","Text":"We can only do this for linear functions."},{"Start":"01:40.175 ","End":"01:41.870","Text":"We also divide by 2."},{"Start":"01:41.870 ","End":"01:44.275","Text":"I put the 2 on the denominator."},{"Start":"01:44.275 ","End":"01:46.460","Text":"With indefinite we had a constant,"},{"Start":"01:46.460 ","End":"01:48.440","Text":"but we won\u0027t need that."},{"Start":"01:48.440 ","End":"01:51.020","Text":"Getting back to here,"},{"Start":"01:51.020 ","End":"01:59.825","Text":"what we get is the limit as b goes to infinity of this thing."},{"Start":"01:59.825 ","End":"02:03.245","Text":"Let\u0027s see, on the denominator I have 4/3."},{"Start":"02:03.245 ","End":"02:06.220","Text":"On the numerator becomes 3/4."},{"Start":"02:06.220 ","End":"02:09.185","Text":"You know what?"},{"Start":"02:09.185 ","End":"02:13.250","Text":"I can put this in front of the integral sign wait while I switch them around."},{"Start":"02:13.250 ","End":"02:16.460","Text":"The constants can come in front of the limit."},{"Start":"02:16.460 ","End":"02:19.745","Text":"What we\u0027re left with is not the integral,"},{"Start":"02:19.745 ","End":"02:22.340","Text":"but just this thing itself,"},{"Start":"02:22.340 ","End":"02:25.465","Text":"which is 2x plus 1 to the 2/3."},{"Start":"02:25.465 ","End":"02:28.740","Text":"I can just write it out here to this 2/3."},{"Start":"02:28.740 ","End":"02:33.530","Text":"All this has got to be taken between 2 and b,"},{"Start":"02:33.530 ","End":"02:34.940","Text":"which means I substitute b,"},{"Start":"02:34.940 ","End":"02:37.400","Text":"substitute 2 and subtract."},{"Start":"02:37.400 ","End":"02:42.870","Text":"What I get is 3/4 of something."},{"Start":"02:42.870 ","End":"02:45.150","Text":"Let\u0027s see if I put b in here,"},{"Start":"02:45.150 ","End":"02:51.825","Text":"I get 2b plus 1 to the power of 2/3 minus,"},{"Start":"02:51.825 ","End":"02:54.450","Text":"let\u0027s see if I put 2 in here."},{"Start":"02:54.450 ","End":"03:00.120","Text":"It\u0027s twice 2 plus 1 is 5 2/3."},{"Start":"03:00.120 ","End":"03:02.250","Text":"That\u0027s just some number."},{"Start":"03:02.250 ","End":"03:05.010","Text":"I forgot to write the limit."},{"Start":"03:05.010 ","End":"03:07.260","Text":"Just let me correct that."},{"Start":"03:07.260 ","End":"03:11.870","Text":"Its limit as b goes to infinity."},{"Start":"03:11.870 ","End":"03:15.320","Text":"Now really what concerns us is this thing."},{"Start":"03:15.320 ","End":"03:16.790","Text":"If b goes to infinity,"},{"Start":"03:16.790 ","End":"03:19.025","Text":"2/3 of 2b plus 1."},{"Start":"03:19.025 ","End":"03:27.830","Text":"Basically what I end up with is 3/4 times 2b plus 1 is also infinity."},{"Start":"03:27.830 ","End":"03:32.890","Text":"Infinity to the 2/3 minus 5 to the 2/3."},{"Start":"03:32.890 ","End":"03:37.730","Text":"The infinity to the power of 2/3 is also infinity."},{"Start":"03:37.730 ","End":"03:42.500","Text":"Infinity minus a constant is infinity times positive constant is still infinity,"},{"Start":"03:42.500 ","End":"03:49.655","Text":"so it\u0027s infinity, which means that our original integral diverges."},{"Start":"03:49.655 ","End":"03:51.905","Text":"This 1 diverges."},{"Start":"03:51.905 ","End":"03:54.574","Text":"We can\u0027t determine its value."},{"Start":"03:54.574 ","End":"03:56.270","Text":"We could say it\u0027s infinity,"},{"Start":"03:56.270 ","End":"04:00.960","Text":"but that counts as diverges. We\u0027re done."}],"ID":4574},{"Watched":false,"Name":"Exercise 5","Duration":"2m 46s","ChapterTopicVideoID":4566,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.785","Text":"Here we\u0027re given an integral."},{"Start":"00:01.785 ","End":"00:05.280","Text":"You have to decide if it diverges or converges,"},{"Start":"00:05.280 ","End":"00:07.710","Text":"in which case we compute it."},{"Start":"00:07.710 ","End":"00:10.515","Text":"I\u0027ve copied it over here."},{"Start":"00:10.515 ","End":"00:14.685","Text":"What I\u0027m going to do is interpret the infinity."},{"Start":"00:14.685 ","End":"00:16.640","Text":"This means it\u0027s an improper integral,"},{"Start":"00:16.640 ","End":"00:19.260","Text":"and what we do is instead of the infinity,"},{"Start":"00:19.260 ","End":"00:22.625","Text":"we put another letter usually b,"},{"Start":"00:22.625 ","End":"00:29.810","Text":"and then I take the limit as b goes to infinity of the same thing."},{"Start":"00:29.810 ","End":"00:32.180","Text":"That\u0027s how we interpret the infinity."},{"Start":"00:32.180 ","End":"00:34.985","Text":"Just take b goes to infinity."},{"Start":"00:34.985 ","End":"00:44.040","Text":"Now, 1 over e^x will be more convenient to write as e^minus x dx."},{"Start":"00:44.440 ","End":"00:49.430","Text":"This integral is an easy one because the integral of"},{"Start":"00:49.430 ","End":"00:53.915","Text":"e^minus x is just e^minus x,"},{"Start":"00:53.915 ","End":"00:56.270","Text":"but we have to divide by minus 1,"},{"Start":"00:56.270 ","End":"00:59.450","Text":"and the minus"},{"Start":"00:59.450 ","End":"01:02.420","Text":"I can pull all the way in front of the integral and in front to the limit,"},{"Start":"01:02.420 ","End":"01:07.505","Text":"so we get minus the limit as b goes to infinity,"},{"Start":"01:07.505 ","End":"01:11.200","Text":"of e^minus x."},{"Start":"01:11.200 ","End":"01:14.320","Text":"Again, just do it at the side, what I just did here."},{"Start":"01:14.320 ","End":"01:19.265","Text":"I just did the indefinite integral of e^minus x dx"},{"Start":"01:19.265 ","End":"01:25.249","Text":"was equal to e^minus x over minus 1,"},{"Start":"01:25.249 ","End":"01:28.205","Text":"which is just minus e^minus x."},{"Start":"01:28.205 ","End":"01:32.485","Text":"That\u0027s all I did here, but I pulled the minus in front of the limit."},{"Start":"01:32.485 ","End":"01:34.920","Text":"I hope that\u0027s clear."},{"Start":"01:34.920 ","End":"01:37.110","Text":"It was a definite integral."},{"Start":"01:37.110 ","End":"01:41.600","Text":"This thing is taken between the limits of 0 and b,"},{"Start":"01:41.600 ","End":"01:44.045","Text":"by which I mean I put in b,"},{"Start":"01:44.045 ","End":"01:46.715","Text":"put in 0 and subtract the two results."},{"Start":"01:46.715 ","End":"01:48.544","Text":"From here, I get"},{"Start":"01:48.544 ","End":"01:57.335","Text":"minus the limit as b goes to infinity, square brackets here."},{"Start":"01:57.335 ","End":"02:02.575","Text":"Firstly, put in b and I\u0027ll get e^minus b."},{"Start":"02:02.575 ","End":"02:06.300","Text":"Put in 0, e^minus 0,"},{"Start":"02:06.300 ","End":"02:10.170","Text":"and what we get is,"},{"Start":"02:10.170 ","End":"02:12.860","Text":"e^minus 0 is 1,"},{"Start":"02:12.860 ","End":"02:17.375","Text":"because e^0, 1 minus e^minus b. I\u0027ve got"},{"Start":"02:17.375 ","End":"02:24.860","Text":"the limit of reversing the order because of the minus 1 minus e^minus b,"},{"Start":"02:24.860 ","End":"02:28.130","Text":"b goes to infinity."},{"Start":"02:28.130 ","End":"02:30.050","Text":"What I get ultimately,"},{"Start":"02:30.050 ","End":"02:35.120","Text":"almost ultimately, is 1 minus e^minus infinity."},{"Start":"02:35.120 ","End":"02:40.350","Text":"Now e^minus infinity is well-known to be 0."},{"Start":"02:40.470 ","End":"02:44.435","Text":"The final answer is just equal to 1,"},{"Start":"02:44.435 ","End":"02:46.860","Text":"and that\u0027s it. We\u0027re done."}],"ID":4575},{"Watched":false,"Name":"Exercise 6","Duration":"2m 17s","ChapterTopicVideoID":4567,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"Here we have to compute the following integral if it"},{"Start":"00:02.940 ","End":"00:06.045","Text":"converges or to see if it may be diverges."},{"Start":"00:06.045 ","End":"00:08.385","Text":"I\u0027ve copied it over here."},{"Start":"00:08.385 ","End":"00:11.910","Text":"This is an improper definite integral,"},{"Start":"00:11.910 ","End":"00:14.460","Text":"the improper because of the minus infinity."},{"Start":"00:14.460 ","End":"00:19.080","Text":"The standard method of doing this is to take the integral and"},{"Start":"00:19.080 ","End":"00:23.715","Text":"instead of the minus infinity to put some letter which will go to minus infinity."},{"Start":"00:23.715 ","End":"00:27.090","Text":"I\u0027m calling this a, this is still 0,"},{"Start":"00:27.090 ","End":"00:32.595","Text":"and I take the limit as a goes to minus infinity."},{"Start":"00:32.595 ","End":"00:35.190","Text":"This minus infinity really is just a symbol of"},{"Start":"00:35.190 ","End":"00:38.475","Text":"something that goes to minus infinity in the limit."},{"Start":"00:38.475 ","End":"00:40.170","Text":"Inside it\u0027s the same thing,"},{"Start":"00:40.170 ","End":"00:44.955","Text":"e to the 4x dx and we\u0027re going to continue."},{"Start":"00:44.955 ","End":"00:47.480","Text":"But to do this definite integral,"},{"Start":"00:47.480 ","End":"00:51.005","Text":"we need the indefinite integral, so why don\u0027t I just do that quickly at the side?"},{"Start":"00:51.005 ","End":"00:54.990","Text":"The indefinite integral e to"},{"Start":"00:54.990 ","End":"01:02.550","Text":"the 4x dx is going to equal, let\u0027s see, e to the 4x would normally be just e to the 4x,"},{"Start":"01:02.550 ","End":"01:04.390","Text":"but because it\u0027s 4x,"},{"Start":"01:04.390 ","End":"01:06.140","Text":"which is a linear function of x,"},{"Start":"01:06.140 ","End":"01:08.150","Text":"we divide by the inner derivative,"},{"Start":"01:08.150 ","End":"01:11.300","Text":"which is 4 so it\u0027s e to the 4 x over 4."},{"Start":"01:11.300 ","End":"01:13.400","Text":"If it\u0027s indefinite, we add a constant,"},{"Start":"01:13.400 ","End":"01:16.970","Text":"but we don\u0027t need this constant when we go and do the definite integral."},{"Start":"01:16.970 ","End":"01:22.340","Text":"What we get is the limit as a goes to minus infinity."},{"Start":"01:22.340 ","End":"01:26.870","Text":"Now this integral becomes just e to the 4x over"},{"Start":"01:26.870 ","End":"01:32.025","Text":"4 between 0 at the top and a at the bottom."},{"Start":"01:32.025 ","End":"01:36.830","Text":"Which means we plug in each of these and subtract this 1 from this 1."},{"Start":"01:36.830 ","End":"01:39.760","Text":"What we get is the limit."},{"Start":"01:39.760 ","End":"01:43.490","Text":"Again, a goes to minus infinity of something minus something."},{"Start":"01:43.490 ","End":"01:47.735","Text":"I put in 0 e^4 times 0 is e^0 is"},{"Start":"01:47.735 ","End":"01:57.560","Text":"1/4 minus 1/4 of e^4a."},{"Start":"01:57.560 ","End":"02:00.680","Text":"Now, if a goes to minus infinity,"},{"Start":"02:00.680 ","End":"02:07.475","Text":"what I get is 1/4 minus 1/4 e^4 times minus infinity,"},{"Start":"02:07.475 ","End":"02:09.690","Text":"that\u0027s still minus infinity."},{"Start":"02:09.690 ","End":"02:13.280","Text":"Since e to the minus infinity is 0,"},{"Start":"02:13.280 ","End":"02:18.540","Text":"I\u0027m just left with 1/4 and that is our answer."}],"ID":4576},{"Watched":false,"Name":"Exercise 7","Duration":"3m 21s","ChapterTopicVideoID":4568,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.155","Text":"Here we have to check if this integral converges or diverges."},{"Start":"00:04.155 ","End":"00:08.925","Text":"If it converges, we have to evaluate it. I just copied it here."},{"Start":"00:08.925 ","End":"00:11.130","Text":"The reason I\u0027m talking about converging or"},{"Start":"00:11.130 ","End":"00:13.470","Text":"diverging is that this is an improper integral."},{"Start":"00:13.470 ","End":"00:15.705","Text":"It\u0027s improper because of the infinity."},{"Start":"00:15.705 ","End":"00:19.140","Text":"The infinity is some shorthand code for"},{"Start":"00:19.140 ","End":"00:23.505","Text":"writing the limit when the upper bound goes to infinity."},{"Start":"00:23.505 ","End":"00:26.085","Text":"What I mean is this is equal to limit,"},{"Start":"00:26.085 ","End":"00:27.480","Text":"to choose some letter,"},{"Start":"00:27.480 ","End":"00:28.935","Text":"let\u0027s call it B."},{"Start":"00:28.935 ","End":"00:30.795","Text":"From minus 1 to B,"},{"Start":"00:30.795 ","End":"00:32.985","Text":"but B goes to infinity."},{"Start":"00:32.985 ","End":"00:35.370","Text":"That\u0027s how to interpret the infinity here."},{"Start":"00:35.370 ","End":"00:39.505","Text":"The same X over 1 plus X squared dx."},{"Start":"00:39.505 ","End":"00:42.350","Text":"Now I\u0027m going to continue, but I need the indefinite integral,"},{"Start":"00:42.350 ","End":"00:45.625","Text":"the anti-derivative, so I\u0027ll do that at the side."},{"Start":"00:45.625 ","End":"00:53.050","Text":"The integral of X over 1 plus X squared dx."},{"Start":"00:53.050 ","End":"00:57.170","Text":"I almost have the derivative of the denominator and the numerator,"},{"Start":"00:57.170 ","End":"00:58.840","Text":"if it was 2x here,"},{"Start":"00:58.840 ","End":"01:02.130","Text":"well, why not make it 2x?"},{"Start":"01:02.130 ","End":"01:06.300","Text":"Let\u0027s try 2x over 1 plus X squared dx."},{"Start":"01:06.300 ","End":"01:09.210","Text":"But you say, \"Hey, I can\u0027t just add a 2,\" and you\u0027re right,"},{"Start":"01:09.210 ","End":"01:10.500","Text":"we have to compensate."},{"Start":"01:10.500 ","End":"01:12.510","Text":"Let\u0027s write a 1/2 out here."},{"Start":"01:12.510 ","End":"01:17.125","Text":"Now because it\u0027s of the form F prime over F,"},{"Start":"01:17.125 ","End":"01:21.155","Text":"we know that the integral of this is a natural log of the denominator."},{"Start":"01:21.155 ","End":"01:25.025","Text":"We get 1.5 natural log,"},{"Start":"01:25.025 ","End":"01:27.910","Text":"actually need an absolute value,"},{"Start":"01:27.910 ","End":"01:32.310","Text":"of 1 plus X squared."},{"Start":"01:32.310 ","End":"01:35.720","Text":"Because it\u0027s indefinite, we add a constant,"},{"Start":"01:35.720 ","End":"01:39.020","Text":"but we don\u0027t need that constant when we go back here."},{"Start":"01:39.020 ","End":"01:41.410","Text":"What we get is the limit,"},{"Start":"01:41.410 ","End":"01:44.059","Text":"as B goes to infinity,"},{"Start":"01:44.059 ","End":"01:46.235","Text":"we\u0027ve got the integral which is this,"},{"Start":"01:46.235 ","End":"01:47.390","Text":"we\u0027ll put it in brackets,"},{"Start":"01:47.390 ","End":"01:53.060","Text":"1.5 natural log of 1 plus X squared,"},{"Start":"01:53.060 ","End":"01:56.445","Text":"but between minus 1 and B."},{"Start":"01:56.445 ","End":"01:59.594","Text":"We substitute this, substitute this, and subtract."},{"Start":"01:59.594 ","End":"02:03.705","Text":"What we get is, let\u0027s say limits,"},{"Start":"02:03.705 ","End":"02:11.315","Text":"put in B and we have 1.5 natural log of 1 plus B squared."},{"Start":"02:11.315 ","End":"02:14.405","Text":"It\u0027s positive, I don\u0027t need the absolute value."},{"Start":"02:14.405 ","End":"02:16.640","Text":"In fact, it\u0027s always positive,"},{"Start":"02:16.640 ","End":"02:18.115","Text":"so we don\u0027t need the absolute value."},{"Start":"02:18.115 ","End":"02:21.670","Text":"Put in minus 1, minus 1 squared is 1,"},{"Start":"02:21.670 ","End":"02:24.340","Text":"1 plus, let\u0027s see with my calculator,"},{"Start":"02:24.340 ","End":"02:26.155","Text":"1 plus 1 is 2,"},{"Start":"02:26.155 ","End":"02:32.450","Text":"and then we have minus 1.5 natural log of 2."},{"Start":"02:32.450 ","End":"02:34.160","Text":"How do we proceed?"},{"Start":"02:34.160 ","End":"02:35.360","Text":"Just in a way,"},{"Start":"02:35.360 ","End":"02:37.850","Text":"substitute infinity as if it was a number."},{"Start":"02:37.850 ","End":"02:41.290","Text":"We have 1.5 natural log."},{"Start":"02:41.290 ","End":"02:43.830","Text":"Now B is infinity,"},{"Start":"02:43.830 ","End":"02:47.760","Text":"so B squared is also infinity and if I add 1, it\u0027s still infinity."},{"Start":"02:47.760 ","End":"02:50.550","Text":"It\u0027s a natural log of infinity."},{"Start":"02:50.550 ","End":"02:52.800","Text":"This is just a constant,"},{"Start":"02:52.800 ","End":"02:55.575","Text":"it\u0027s 1/2 natural log of 2, it\u0027s a constant."},{"Start":"02:55.575 ","End":"03:01.580","Text":"But the natural log of infinity is also infinity."},{"Start":"03:01.580 ","End":"03:04.245","Text":"A 1/2 of infinity minus a constant;"},{"Start":"03:04.245 ","End":"03:07.005","Text":"in short, the answer is infinity,"},{"Start":"03:07.005 ","End":"03:08.945","Text":"and when the answer is infinity,"},{"Start":"03:08.945 ","End":"03:13.045","Text":"it means that my original integral diverges."},{"Start":"03:13.045 ","End":"03:15.780","Text":"There is no evaluation, no value."},{"Start":"03:15.780 ","End":"03:18.660","Text":"Infinity is not a value, it\u0027s a symbol."},{"Start":"03:18.660 ","End":"03:21.760","Text":"That\u0027s it. We are done."}],"ID":4577},{"Watched":false,"Name":"Exercise 8","Duration":"3m 50s","ChapterTopicVideoID":4569,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"Once again, we have an improper integral which you can see because of the infinity."},{"Start":"00:04.440 ","End":"00:07.950","Text":"We have to decide if it converges or diverges."},{"Start":"00:07.950 ","End":"00:11.085","Text":"If it converges, it means it has a value and we\u0027ve got to find it."},{"Start":"00:11.085 ","End":"00:16.815","Text":"Let me just copy that, and the way to deal with this infinity is the following."},{"Start":"00:16.815 ","End":"00:19.875","Text":"We imagine it\u0027s not an infinity,"},{"Start":"00:19.875 ","End":"00:21.510","Text":"but something going to infinity."},{"Start":"00:21.510 ","End":"00:25.125","Text":"Let\u0027s call it b, so we have from 0 to b,"},{"Start":"00:25.125 ","End":"00:32.820","Text":"b goes to infinity means we take the limit as b tends to infinity of the same thing,"},{"Start":"00:32.820 ","End":"00:36.400","Text":"xe to the minus x squared dx."},{"Start":"00:36.400 ","End":"00:39.035","Text":"Now to evaluate this definite integral,"},{"Start":"00:39.035 ","End":"00:41.540","Text":"I should first do the indefinite integral,"},{"Start":"00:41.540 ","End":"00:44.390","Text":"the anti-derivative and let me do that at the side."},{"Start":"00:44.390 ","End":"00:50.750","Text":"Let\u0027s see, the integral indefinite of xe to the minus x squared."},{"Start":"00:50.750 ","End":"00:52.670","Text":"There\u0027s more than 1 way to do this,"},{"Start":"00:52.670 ","End":"00:55.055","Text":"and I think I\u0027ll do it with a substitution."},{"Start":"00:55.055 ","End":"00:58.520","Text":"Because I see that the derivative of x squared is x,"},{"Start":"00:58.520 ","End":"01:03.210","Text":"I\u0027m going to try to substitute t equals,"},{"Start":"01:03.210 ","End":"01:06.320","Text":"let\u0027s go for the whole minus x squared."},{"Start":"01:06.320 ","End":"01:09.010","Text":"That will probably go better."},{"Start":"01:09.010 ","End":"01:14.780","Text":"Now dt would equal minus 2xdx."},{"Start":"01:15.030 ","End":"01:23.290","Text":"But if I want dx, this will be dt over minus 2x."},{"Start":"01:23.290 ","End":"01:25.175","Text":"Now if I put all this in here,"},{"Start":"01:25.175 ","End":"01:30.570","Text":"what I get is the integral xe to the power of,"},{"Start":"01:30.570 ","End":"01:33.125","Text":"now minus x squared is t,"},{"Start":"01:33.125 ","End":"01:40.425","Text":"and dx is dt over minus 2x."},{"Start":"01:40.425 ","End":"01:44.870","Text":"Something cancels, the x cancels and that\u0027s good because we want"},{"Start":"01:44.870 ","End":"01:49.765","Text":"to be left with just t. This is a constant,"},{"Start":"01:49.765 ","End":"01:52.520","Text":"so it will come in front of the integration sign."},{"Start":"01:52.520 ","End":"01:59.600","Text":"Basically I get minus 1/2 the integral of e^t dt,"},{"Start":"01:59.600 ","End":"02:04.860","Text":"which is just minus 1/2 e^t,"},{"Start":"02:04.860 ","End":"02:06.460","Text":"here we could put a plus C,"},{"Start":"02:06.460 ","End":"02:09.905","Text":"no need to, because we\u0027re going to do definite integrals."},{"Start":"02:09.905 ","End":"02:16.070","Text":"Getting back here, we have the limit as b goes to infinity."},{"Start":"02:16.070 ","End":"02:18.365","Text":"The integral we\u0027ve already done."},{"Start":"02:18.365 ","End":"02:20.810","Text":"This is 1/2 here, it\u0027s a bit small."},{"Start":"02:20.810 ","End":"02:25.610","Text":"Silly me, I forgot to switch back from t to x. I can\u0027t leave it like this."},{"Start":"02:25.610 ","End":"02:34.620","Text":"This has to equal minus 1/2 e to the minus x squared,"},{"Start":"02:34.620 ","End":"02:39.245","Text":"I\u0027m leaving out the C. This is what I want to put in here,"},{"Start":"02:39.245 ","End":"02:44.150","Text":"which means that I have minus 1/2 e to"},{"Start":"02:44.150 ","End":"02:50.855","Text":"the minus x squared between 0 and b."},{"Start":"02:50.855 ","End":"02:53.425","Text":"Let\u0027s see what this gives us."},{"Start":"02:53.425 ","End":"03:01.335","Text":"The minus 1/2 can come out front and I\u0027ve got the limit, b goes to infinity."},{"Start":"03:01.335 ","End":"03:04.460","Text":"Put first of all the b in here,"},{"Start":"03:04.460 ","End":"03:06.410","Text":"we just have left with the e to the power of,"},{"Start":"03:06.410 ","End":"03:09.575","Text":"so it\u0027s e to the minus b squared."},{"Start":"03:09.575 ","End":"03:13.524","Text":"Then I have the 0, e to the minus 0 squared,"},{"Start":"03:13.524 ","End":"03:16.395","Text":"I\u0027ll write it, but it\u0027s just equal to 1,"},{"Start":"03:16.395 ","End":"03:23.375","Text":"so this is equal to minus 1/2 the limit b goes to infinity,"},{"Start":"03:23.375 ","End":"03:27.580","Text":"e to the minus b squared minus 1."},{"Start":"03:27.580 ","End":"03:30.150","Text":"When b goes to infinity,"},{"Start":"03:30.150 ","End":"03:31.935","Text":"so does b squared."},{"Start":"03:31.935 ","End":"03:36.904","Text":"What we get is e to the minus infinity minus 1."},{"Start":"03:36.904 ","End":"03:40.610","Text":"Now, e to the minus infinity is known to be 0."},{"Start":"03:40.610 ","End":"03:45.185","Text":"All I\u0027m left with is minus 1/2 times minus 1,"},{"Start":"03:45.185 ","End":"03:49.595","Text":"which is equal to plus 1/2 and that\u0027s it."},{"Start":"03:49.595 ","End":"03:51.660","Text":"We are done."}],"ID":4578},{"Watched":false,"Name":"Exercise 9","Duration":"3m 52s","ChapterTopicVideoID":4570,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"Here again, we have an improper integral because of the infinity here."},{"Start":"00:05.040 ","End":"00:09.105","Text":"If it\u0027s improper, it could converge or diverge."},{"Start":"00:09.105 ","End":"00:11.565","Text":"If it converges, we have to evaluate it."},{"Start":"00:11.565 ","End":"00:13.455","Text":"I\u0027ve copied it over here."},{"Start":"00:13.455 ","End":"00:15.390","Text":"Let\u0027s see from e to infinity."},{"Start":"00:15.390 ","End":"00:17.039","Text":"Now, what do we do with this infinity?"},{"Start":"00:17.039 ","End":"00:20.550","Text":"The standard thing to do is set to infinity,"},{"Start":"00:20.550 ","End":"00:24.720","Text":"replace it with something that tends to infinity like the letter b,"},{"Start":"00:24.720 ","End":"00:28.980","Text":"and then take the limit as b goes to infinity."},{"Start":"00:28.980 ","End":"00:32.279","Text":"Everything else is the same, e here,"},{"Start":"00:32.279 ","End":"00:38.930","Text":"1 over x natural log to the fourth of x dx."},{"Start":"00:38.930 ","End":"00:41.780","Text":"Before I do the definite integral,"},{"Start":"00:41.780 ","End":"00:45.215","Text":"I want to do the indefinite integral, the anti-derivative."},{"Start":"00:45.215 ","End":"00:47.165","Text":"I\u0027ll do it at the side here."},{"Start":"00:47.165 ","End":"00:48.440","Text":"What I want to know is,"},{"Start":"00:48.440 ","End":"00:52.935","Text":"what is the integral of dx?"},{"Start":"00:52.935 ","End":"00:54.980","Text":"Just to save space, I\u0027ll put it on the top,"},{"Start":"00:54.980 ","End":"01:00.130","Text":"over x, natural log to the fourth of x."},{"Start":"01:00.130 ","End":"01:05.945","Text":"Now, I look at this and what comes to mind is a substitution."},{"Start":"01:05.945 ","End":"01:09.529","Text":"If I replace natural log of x with t,"},{"Start":"01:09.529 ","End":"01:17.345","Text":"that\u0027s going to work out fine because the derivative dt is just 1 over x dx."},{"Start":"01:17.345 ","End":"01:19.245","Text":"I\u0027ll write that down,"},{"Start":"01:19.245 ","End":"01:21.890","Text":"dt is 1 over x dx."},{"Start":"01:21.890 ","End":"01:24.350","Text":"Sometimes I write that as dx over x."},{"Start":"01:24.350 ","End":"01:26.960","Text":"Now, this will substitute very nicely."},{"Start":"01:26.960 ","End":"01:28.190","Text":"The dx over x,"},{"Start":"01:28.190 ","End":"01:31.270","Text":"I can see already it\u0027s just equal to dt,"},{"Start":"01:31.270 ","End":"01:36.260","Text":"and the natural log of x is t over t to the fourth."},{"Start":"01:36.260 ","End":"01:38.030","Text":"What it really equals,"},{"Start":"01:38.030 ","End":"01:42.065","Text":"if I put it in exponent form not in the denominator,"},{"Start":"01:42.065 ","End":"01:47.815","Text":"it\u0027s the integral of t to the minus 4dt."},{"Start":"01:47.815 ","End":"01:49.909","Text":"I just didn\u0027t want it in the denominator,"},{"Start":"01:49.909 ","End":"01:51.740","Text":"because I know how to do exponents."},{"Start":"01:51.740 ","End":"01:54.935","Text":"I raise the exponent by 1 and then divide by it."},{"Start":"01:54.935 ","End":"02:00.245","Text":"This is just t to the minus 3 over minus 3."},{"Start":"02:00.245 ","End":"02:02.660","Text":"You can put the plus c if you\u0027re used to it."},{"Start":"02:02.660 ","End":"02:04.535","Text":"Not necessary here."},{"Start":"02:04.535 ","End":"02:07.385","Text":"Now going back over here,"},{"Start":"02:07.385 ","End":"02:10.790","Text":"we now have the indefinite integral so we can continue."},{"Start":"02:10.790 ","End":"02:15.080","Text":"This is the limit as b goes to infinity,"},{"Start":"02:15.080 ","End":"02:19.860","Text":"t to the minus 3 over minus 3,"},{"Start":"02:19.860 ","End":"02:24.060","Text":"taken between e and b."},{"Start":"02:24.060 ","End":"02:29.790","Text":"This equals limit b goes to infinity."},{"Start":"02:29.790 ","End":"02:31.310","Text":"Now, what do I do with this?"},{"Start":"02:31.310 ","End":"02:34.930","Text":"I put b in and I put e in and I subtract."},{"Start":"02:34.930 ","End":"02:41.505","Text":"If I put b, I\u0027ve got b to the minus 3 over minus 3,"},{"Start":"02:41.505 ","End":"02:43.325","Text":"and if I put e in,"},{"Start":"02:43.325 ","End":"02:48.185","Text":"I\u0027ve got e to the minus 3 over minus 3."},{"Start":"02:48.185 ","End":"02:52.100","Text":"Now, the thing is, what happens when b goes to infinity?"},{"Start":"02:52.100 ","End":"02:56.810","Text":"The important term I would say is this 1."},{"Start":"02:56.810 ","End":"02:58.654","Text":"This is where it all happens."},{"Start":"02:58.654 ","End":"03:00.575","Text":"Because this bit here,"},{"Start":"03:00.575 ","End":"03:04.310","Text":"and I\u0027ll just do this at the side somewhere, like over here,"},{"Start":"03:04.310 ","End":"03:11.690","Text":"b to the minus 3 is the same as 1 over b cubed."},{"Start":"03:11.690 ","End":"03:14.465","Text":"Now, if b goes to infinity,"},{"Start":"03:14.465 ","End":"03:17.555","Text":"then b cubed also goes to infinity,"},{"Start":"03:17.555 ","End":"03:21.535","Text":"so 1 over b cubed is 1 over infinity,"},{"Start":"03:21.535 ","End":"03:25.210","Text":"which is just 0."},{"Start":"03:25.490 ","End":"03:27.590","Text":"What I get here,"},{"Start":"03:27.590 ","End":"03:33.095","Text":"if this thing is 0, this minus with this minus goes."},{"Start":"03:33.095 ","End":"03:39.370","Text":"What I get is e to the minus 3 over 3,"},{"Start":"03:39.370 ","End":"03:42.935","Text":"because the minus with the minus cancels."},{"Start":"03:42.935 ","End":"03:45.230","Text":"If you\u0027d like to put it in the denominator,"},{"Start":"03:45.230 ","End":"03:53.490","Text":"we could also say this is 1 over 3e cubed. We are done."}],"ID":4579},{"Watched":false,"Name":"Exercise 10","Duration":"4m 1s","ChapterTopicVideoID":4571,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"What I have here is an improper integral to compute because of the infinity here."},{"Start":"00:05.550 ","End":"00:07.440","Text":"That\u0027s why it\u0027s improper."},{"Start":"00:07.440 ","End":"00:10.425","Text":"In that case, it could converge or diverge."},{"Start":"00:10.425 ","End":"00:13.140","Text":"If it diverges, it won\u0027t have a value."},{"Start":"00:13.140 ","End":"00:15.555","Text":"I\u0027ve copied it here."},{"Start":"00:15.555 ","End":"00:18.180","Text":"How do we deal with this infinity?"},{"Start":"00:18.180 ","End":"00:23.190","Text":"The standard thing to do is to take something that turns to infinity."},{"Start":"00:23.190 ","End":"00:28.320","Text":"Use the letter b, and I\u0027m going to let b turn to infinity by means of the limit,"},{"Start":"00:28.320 ","End":"00:30.570","Text":"and everything else remains the same."},{"Start":"00:30.570 ","End":"00:37.210","Text":"1 over x square root of natural log of x dx."},{"Start":"00:37.210 ","End":"00:40.055","Text":"I computed for a parameter b,"},{"Start":"00:40.055 ","End":"00:42.755","Text":"like a number, and that b turned to infinity."},{"Start":"00:42.755 ","End":"00:45.560","Text":"I need first of all to know the indefinite integral,"},{"Start":"00:45.560 ","End":"00:47.880","Text":"the anti-derivative of this thing,"},{"Start":"00:47.880 ","End":"00:49.485","Text":"and I\u0027ll do that at the side."},{"Start":"00:49.485 ","End":"00:58.895","Text":"The question is, what is the integral of 1 over x square root of the natural log of x dx."},{"Start":"00:58.895 ","End":"01:00.470","Text":"Several ways to do this."},{"Start":"01:00.470 ","End":"01:02.165","Text":"I recommend a substitution."},{"Start":"01:02.165 ","End":"01:05.840","Text":"I would let t equal either natural log of x"},{"Start":"01:05.840 ","End":"01:07.700","Text":"or square root of natural log of x."},{"Start":"01:07.700 ","End":"01:08.570","Text":"Both will work."},{"Start":"01:08.570 ","End":"01:11.755","Text":"I\u0027ll just go with natural log of x."},{"Start":"01:11.755 ","End":"01:19.280","Text":"Then dt is equal to 1 over x dx."},{"Start":"01:19.340 ","End":"01:22.140","Text":"I already have the 1 over x here,"},{"Start":"01:22.140 ","End":"01:23.635","Text":"so after the substitution,"},{"Start":"01:23.635 ","End":"01:30.220","Text":"I get 1 over x dx is dt and the square root"},{"Start":"01:30.220 ","End":"01:33.820","Text":"of natural log of x is just the square root of t."},{"Start":"01:33.820 ","End":"01:37.270","Text":"This we\u0027ve done plenty of times."},{"Start":"01:37.270 ","End":"01:41.290","Text":"What we usually do is just put a 2 here and"},{"Start":"01:41.290 ","End":"01:44.050","Text":"a 2 here where these cancel each other"},{"Start":"01:44.050 ","End":"01:46.540","Text":"because this is an immediate integral."},{"Start":"01:46.540 ","End":"01:50.770","Text":"Because this just equals twice square root of t."},{"Start":"01:50.770 ","End":"01:55.525","Text":"Remember, the derivative of root t is 1 over twice root t plus a constant."},{"Start":"01:55.525 ","End":"01:57.310","Text":"At the end, you have to switch back,"},{"Start":"01:57.310 ","End":"01:59.800","Text":"so we have this is equal to"},{"Start":"01:59.800 ","End":"02:08.960","Text":"twice natural log of the square root of t plus the constant if you like."},{"Start":"02:08.960 ","End":"02:16.415","Text":"Now, I can go back here and continue the story and say this is equal to the limit,"},{"Start":"02:16.415 ","End":"02:24.950","Text":"same limit of twice natural log of square root of t."},{"Start":"02:24.950 ","End":"02:29.770","Text":"You know what, I\u0027m tempted to use a trick, well, dt,"},{"Start":"02:29.770 ","End":"02:36.065","Text":"because the properties of the logarithm are very well known."},{"Start":"02:36.065 ","End":"02:39.545","Text":"I mean, I\u0027ll scroll down a bit to get more space."},{"Start":"02:39.545 ","End":"02:41.420","Text":"What is the square root of t?"},{"Start":"02:41.420 ","End":"02:43.085","Text":"I\u0027m going to continue here."},{"Start":"02:43.085 ","End":"02:48.200","Text":"The square root of t is t to the power of 1/2."},{"Start":"02:48.200 ","End":"02:50.780","Text":"Using the rules of logarithms,"},{"Start":"02:50.780 ","End":"02:53.075","Text":"when you have a power,"},{"Start":"02:53.075 ","End":"02:56.960","Text":"and you take the logarithm, the power comes out in front as a multiplier."},{"Start":"02:56.960 ","End":"02:59.615","Text":"What I get is the 2-stage from before and I get"},{"Start":"02:59.615 ","End":"03:05.580","Text":"1/2 natural log of t and the 2 with the 2 cancel."},{"Start":"03:05.580 ","End":"03:09.590","Text":"Really, the answer to the integral was natural log of t."},{"Start":"03:09.590 ","End":"03:12.830","Text":"What I\u0027m gonna do is I\u0027m going to replace"},{"Start":"03:12.830 ","End":"03:16.700","Text":"this thing here and just write natural log of t."},{"Start":"03:16.700 ","End":"03:19.385","Text":"l don\u0027t know why I wrote that dt before."},{"Start":"03:19.385 ","End":"03:24.680","Text":"Anyway, this is taken between 4 and b,"},{"Start":"03:24.680 ","End":"03:26.675","Text":"and then b goes to infinity."},{"Start":"03:26.675 ","End":"03:29.050","Text":"What we get is the limit."},{"Start":"03:29.050 ","End":"03:31.160","Text":"Now, if I put in b,"},{"Start":"03:31.160 ","End":"03:33.065","Text":"I get natural log of b,"},{"Start":"03:33.065 ","End":"03:35.570","Text":"put in 4, I get natural log of 4."},{"Start":"03:35.570 ","End":"03:39.065","Text":"I subtract them, b goes to infinity."},{"Start":"03:39.065 ","End":"03:41.195","Text":"If b goes to infinity,"},{"Start":"03:41.195 ","End":"03:44.345","Text":"natural log of b also goes to infinity,"},{"Start":"03:44.345 ","End":"03:46.880","Text":"and the constant makes no difference."},{"Start":"03:46.880 ","End":"03:50.990","Text":"We really just get infinity minus a constant,"},{"Start":"03:50.990 ","End":"03:52.940","Text":"which is just infinity."},{"Start":"03:52.940 ","End":"03:56.240","Text":"If it\u0027s infinity, then what that says about the"},{"Start":"03:56.240 ","End":"04:00.800","Text":"original integral is that it diverges."},{"Start":"04:00.800 ","End":"04:02.520","Text":"We\u0027re done."}],"ID":4580},{"Watched":false,"Name":"Exercise 11","Duration":"2m ","ChapterTopicVideoID":4572,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.195","Text":"Here we have to decide if this improper integral converges or diverges."},{"Start":"00:06.195 ","End":"00:08.610","Text":"If it converges, then compute it."},{"Start":"00:08.610 ","End":"00:11.955","Text":"It\u0027s improper because of the infinity here."},{"Start":"00:11.955 ","End":"00:15.630","Text":"What we do in the case of infinity is we"},{"Start":"00:15.630 ","End":"00:19.080","Text":"replace it by something that approaches infinity."},{"Start":"00:19.080 ","End":"00:25.335","Text":"I use the letter b and I let the limit as b goes to infinity."},{"Start":"00:25.335 ","End":"00:31.360","Text":"From 0, the same 1 over 1 plus x squared dx."},{"Start":"00:31.720 ","End":"00:36.300","Text":"This happens to be an immediate integral"},{"Start":"00:36.300 ","End":"00:41.085","Text":"because the integral of 1 over 1 plus x squared is the arctangent."},{"Start":"00:41.085 ","End":"00:45.110","Text":"What I get is the limit as b goes to infinity"},{"Start":"00:45.110 ","End":"00:53.405","Text":"of arctangent x between 0 and b."},{"Start":"00:53.405 ","End":"00:56.330","Text":"Which simply means that I have to substitute b,"},{"Start":"00:56.330 ","End":"00:58.825","Text":"substitute 0 and subtract,"},{"Start":"00:58.825 ","End":"01:00.645","Text":"so I get the limit."},{"Start":"01:00.645 ","End":"01:11.265","Text":"Again b goes to infinity of arctangent b minus arctangent 0."},{"Start":"01:11.265 ","End":"01:17.270","Text":"Let\u0027s see. Arctangent of 0 is 0 either on the calculator or just think about it,"},{"Start":"01:17.270 ","End":"01:18.710","Text":"the tangent of 0 is 0,"},{"Start":"01:18.710 ","End":"01:20.660","Text":"so the reverse is true too."},{"Start":"01:20.660 ","End":"01:26.170","Text":"Now the arctangent of b as b goes to infinity,"},{"Start":"01:26.170 ","End":"01:31.590","Text":"I would just call this arctangent of infinity,"},{"Start":"01:31.590 ","End":"01:33.420","Text":"it\u0027s a positive infinity."},{"Start":"01:33.420 ","End":"01:38.020","Text":"The arctangent of infinity does have a value,"},{"Start":"01:38.020 ","End":"01:40.140","Text":"and it\u0027s equal to,"},{"Start":"01:40.140 ","End":"01:41.669","Text":"I think of it in degrees,"},{"Start":"01:41.669 ","End":"01:45.440","Text":"as 90 degrees because when the angle goes to 90 degrees,"},{"Start":"01:45.440 ","End":"01:48.005","Text":"the tangent goes to infinity."},{"Start":"01:48.005 ","End":"01:51.110","Text":"But here we are not using degrees,"},{"Start":"01:51.110 ","End":"01:55.040","Text":"so 90 degrees is Pi over 2."},{"Start":"01:55.040 ","End":"01:57.530","Text":"I\u0027ll just write in brackets 90 degrees if it helps anyone,"},{"Start":"01:57.530 ","End":"02:01.320","Text":"but this is the answer and we\u0027re done."}],"ID":4581},{"Watched":false,"Name":"Exercise 12","Duration":"4m 56s","ChapterTopicVideoID":4573,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.390","Text":"Here we have an improper integral as you can tell by the infinity."},{"Start":"00:04.390 ","End":"00:07.920","Text":"We have to decide if it converges or diverges."},{"Start":"00:07.920 ","End":"00:10.565","Text":"If it converges, we have to compute it."},{"Start":"00:10.565 ","End":"00:12.855","Text":"Let\u0027s see what we have."},{"Start":"00:12.855 ","End":"00:18.780","Text":"The infinity is interpreted by taking something that tends to infinity."},{"Start":"00:18.780 ","End":"00:24.750","Text":"I\u0027ll use the letter b and I\u0027ll take the limit as b goes to infinity,"},{"Start":"00:24.750 ","End":"00:27.055","Text":"so instead of infinity, I have a limit."},{"Start":"00:27.055 ","End":"00:36.305","Text":"Everything else is the same 1 e to the minus root x over root x dx."},{"Start":"00:36.305 ","End":"00:39.470","Text":"Now in order to compute this definite integral,"},{"Start":"00:39.470 ","End":"00:41.420","Text":"I need to know the indefinite integral,"},{"Start":"00:41.420 ","End":"00:44.750","Text":"the anti-derivative of this thing here."},{"Start":"00:44.750 ","End":"00:47.285","Text":"I\u0027ll do that at the side."},{"Start":"00:47.285 ","End":"00:49.085","Text":"Let\u0027s see what we get."},{"Start":"00:49.085 ","End":"00:54.394","Text":"The integral of e to the minus root x"},{"Start":"00:54.394 ","End":"01:01.190","Text":"over root x dx should work out with a simple substitution."},{"Start":"01:01.190 ","End":"01:04.595","Text":"I would say t equals the square root of x,"},{"Start":"01:04.595 ","End":"01:06.280","Text":"or minus the square root of x,"},{"Start":"01:06.280 ","End":"01:09.245","Text":"so e to the minus square root of x, it\u0027s a little work,"},{"Start":"01:09.245 ","End":"01:11.420","Text":"I\u0027ll just take it as the square root of x,"},{"Start":"01:11.420 ","End":"01:12.755","Text":"so I\u0027ll keep it simple,"},{"Start":"01:12.755 ","End":"01:15.590","Text":"t equals the square root of x,"},{"Start":"01:15.590 ","End":"01:20.030","Text":"and so dt, this is almost immediate."},{"Start":"01:20.030 ","End":"01:27.270","Text":"This derivative it\u0027s 1 over twice the square root of x dx."},{"Start":"01:27.280 ","End":"01:29.960","Text":"Let\u0027s see what we get."},{"Start":"01:29.960 ","End":"01:35.155","Text":"Over here, I\u0027m just continuing down here."},{"Start":"01:35.155 ","End":"01:38.515","Text":"I would really like to have a 2 here,"},{"Start":"01:38.515 ","End":"01:43.275","Text":"because then I\u0027d have 1 over 2 square root of x dx."},{"Start":"01:43.275 ","End":"01:45.450","Text":"Let me put a 2 here,"},{"Start":"01:45.450 ","End":"01:47.355","Text":"of course, I have to compensate."},{"Start":"01:47.355 ","End":"01:54.120","Text":"Let\u0027s put a 2 out here also to cancel with the 2."},{"Start":"01:54.120 ","End":"02:00.560","Text":"I think that will work better because now we have twice the integral."},{"Start":"02:00.560 ","End":"02:02.600","Text":"Now square root of x is t,"},{"Start":"02:02.600 ","End":"02:09.890","Text":"so I have e to the minus t and dx over twice the square root of x,"},{"Start":"02:09.890 ","End":"02:12.295","Text":"this thing here is just dt."},{"Start":"02:12.295 ","End":"02:15.185","Text":"That\u0027s much easier to compute."},{"Start":"02:15.185 ","End":"02:20.675","Text":"The integral of this is easily seen to be minus e to the minus t."},{"Start":"02:20.675 ","End":"02:28.220","Text":"Altogether I get minus 2e to the minus t plus a constant if you wish."},{"Start":"02:28.220 ","End":"02:33.815","Text":"Now I can go back here and continue"},{"Start":"02:33.815 ","End":"02:40.910","Text":"and say that this is equal to the limit as b goes to infinity."},{"Start":"02:40.910 ","End":"02:45.095","Text":"Silly me, I have to change back from t to x."},{"Start":"02:45.095 ","End":"02:55.535","Text":"Yes, of course, this is equal to minus 2e to the minus square root of x."},{"Start":"02:55.535 ","End":"02:56.630","Text":"That\u0027s better."},{"Start":"02:56.630 ","End":"03:07.170","Text":"We have minus 2e to the minus square root of x taken between 1 and b."},{"Start":"03:07.240 ","End":"03:09.905","Text":"The little trick that\u0027s useful;"},{"Start":"03:09.905 ","End":"03:11.705","Text":"if you have a minus here,"},{"Start":"03:11.705 ","End":"03:14.620","Text":"all you have to do is subtract the other way around,"},{"Start":"03:14.620 ","End":"03:19.355","Text":"plug in this, plug in that and subtract the bottom minus the top."},{"Start":"03:19.355 ","End":"03:24.755","Text":"If I plug in the 1 and I ignore the minus,"},{"Start":"03:24.755 ","End":"03:29.270","Text":"I\u0027m taking off the minus to a plus and I\u0027m switching the order of these 2."},{"Start":"03:29.270 ","End":"03:32.930","Text":"That\u0027s a standard trick and I recommend it."},{"Start":"03:32.930 ","End":"03:36.050","Text":"We get, I plug in 1,"},{"Start":"03:36.050 ","End":"03:40.685","Text":"it\u0027s 2e to the minus square root of 1,"},{"Start":"03:40.685 ","End":"03:47.905","Text":"and then minus 2e to the minus square root of b,"},{"Start":"03:47.905 ","End":"03:50.640","Text":"but I still have the limit, here it is."},{"Start":"03:50.640 ","End":"03:54.690","Text":"This equals, well,"},{"Start":"03:54.690 ","End":"03:59.680","Text":"this is just e to the minus square root of 1 and e to the minus 1 is 1 over e,"},{"Start":"03:59.680 ","End":"04:02.075","Text":"this part is 2 over e,"},{"Start":"04:02.075 ","End":"04:09.515","Text":"and this is 2 over e to the square root of b."},{"Start":"04:09.515 ","End":"04:16.265","Text":"Again, we need the limit as b goes to infinity."},{"Start":"04:16.265 ","End":"04:17.720","Text":"Let\u0027s see what happens."},{"Start":"04:17.720 ","End":"04:20.695","Text":"When b goes to infinity,"},{"Start":"04:20.695 ","End":"04:25.110","Text":"then the square root of b also goes to infinity,"},{"Start":"04:25.110 ","End":"04:27.055","Text":"and e to the infinity,"},{"Start":"04:27.055 ","End":"04:28.390","Text":"now that\u0027s what we get."},{"Start":"04:28.390 ","End":"04:36.620","Text":"We get 2 over e minus 2 over e to the power of infinity."},{"Start":"04:36.620 ","End":"04:39.965","Text":"Now e to the power infinity is also infinity."},{"Start":"04:39.965 ","End":"04:42.440","Text":"2 over infinity is 0."},{"Start":"04:42.440 ","End":"04:45.990","Text":"This thing altogether goes to 0."},{"Start":"04:45.990 ","End":"04:52.515","Text":"What we\u0027re left with is just 2 over e,"},{"Start":"04:52.515 ","End":"04:56.830","Text":"and that\u0027s the answer and we\u0027re done."}],"ID":4582},{"Watched":false,"Name":"Exercise 13","Duration":"9m 8s","ChapterTopicVideoID":4574,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"Here we have an improper integral because you can see the infinity"},{"Start":"00:03.540 ","End":"00:07.380","Text":"here so it could converge or diverge."},{"Start":"00:07.380 ","End":"00:09.959","Text":"We have to decide which and if it converges,"},{"Start":"00:09.959 ","End":"00:12.550","Text":"we have to compute it also."},{"Start":"00:12.830 ","End":"00:18.765","Text":"What I\u0027m going to do is first of all interpret the infinity."},{"Start":"00:18.765 ","End":"00:23.070","Text":"The usual thing to do is to set an infinity, put some letter,"},{"Start":"00:23.070 ","End":"00:28.450","Text":"usually b, and then let the limit as b goes to infinity."},{"Start":"00:28.450 ","End":"00:32.290","Text":"That replaces the infinity is the limit to infinity."},{"Start":"00:32.510 ","End":"00:35.445","Text":"Here everything is the same,"},{"Start":"00:35.445 ","End":"00:39.715","Text":"0,1 over 1 plus e^x dx."},{"Start":"00:39.715 ","End":"00:42.295","Text":"Now this is just a definite integral."},{"Start":"00:42.295 ","End":"00:44.030","Text":"But to compute the definite integral,"},{"Start":"00:44.030 ","End":"00:47.615","Text":"I need to know the indefinite integral and I\u0027m going to do that at the side."},{"Start":"00:47.615 ","End":"00:56.815","Text":"Let\u0027s see if we can compute what is the integral of 1 over 1 plus e^x dx."},{"Start":"00:56.815 ","End":"01:00.440","Text":"I think this will be done by a substitution."},{"Start":"01:00.440 ","End":"01:06.590","Text":"Let\u0027s try substituting t equals e^x,"},{"Start":"01:06.590 ","End":"01:14.905","Text":"and then dt will be equal to e^x dx."},{"Start":"01:14.905 ","End":"01:23.760","Text":"If we need it, dx is equal to dt of e^x."},{"Start":"01:23.760 ","End":"01:27.030","Text":"If we substitute, let\u0027s see what we get."},{"Start":"01:27.030 ","End":"01:29.720","Text":"I\u0027m first going to substitute the dx."},{"Start":"01:29.720 ","End":"01:33.215","Text":"I can see just looking ahead that this is going to be best."},{"Start":"01:33.215 ","End":"01:35.840","Text":"First of all, I\u0027m continuing down here."},{"Start":"01:35.840 ","End":"01:39.085","Text":"First I\u0027m going to substitute the dx,"},{"Start":"01:39.085 ","End":"01:46.105","Text":"and dx is dt over e^x or over e^x dt."},{"Start":"01:46.105 ","End":"01:49.620","Text":"Now I\u0027m going to use t equals e^x."},{"Start":"01:49.620 ","End":"01:56.084","Text":"I get the integral of 1 over 1 plus t,"},{"Start":"01:56.084 ","End":"02:01.030","Text":"and here times 1 over t, dt."},{"Start":"02:01.100 ","End":"02:06.385","Text":"This looks like a classic case of partial fractions."},{"Start":"02:06.385 ","End":"02:10.105","Text":"I don\u0027t want to waste time with partial fractions."},{"Start":"02:10.105 ","End":"02:14.070","Text":"I think I\u0027ll compute it for you at the side and then get back."},{"Start":"02:14.070 ","End":"02:18.100","Text":"The first I\u0027ll just mention the general form that I\u0027m expecting."},{"Start":"02:18.100 ","End":"02:27.510","Text":"I\u0027m expecting this to be of the form A over 1 plus t plus B over t,"},{"Start":"02:27.510 ","End":"02:29.695","Text":"where A and B are some constants."},{"Start":"02:29.695 ","End":"02:34.090","Text":"What I need to do is tell you what A is and B is."},{"Start":"02:34.090 ","End":"02:35.950","Text":"This is a bit of computation,"},{"Start":"02:35.950 ","End":"02:39.485","Text":"so I\u0027ll just as I said do it for you."},{"Start":"02:39.485 ","End":"02:43.375","Text":"I\u0027m back and I computed that this is the case,"},{"Start":"02:43.375 ","End":"02:46.600","Text":"so what we have is this is minus 1 and 1."},{"Start":"02:46.600 ","End":"02:49.180","Text":"Let me just reverse the order so I\u0027ll have the plus first."},{"Start":"02:49.180 ","End":"02:55.245","Text":"What we get is 1 over t minus from the minus here,"},{"Start":"02:55.245 ","End":"03:01.720","Text":"1 over 1 plus t. You mustn\u0027t forget that we\u0027re in the middle of an integral,"},{"Start":"03:01.720 ","End":"03:09.000","Text":"so this is the integral of this dt and this is going to equal."},{"Start":"03:09.000 ","End":"03:17.550","Text":"The integral of this is natural log of t minus natural log of 1"},{"Start":"03:17.550 ","End":"03:21.380","Text":"plus t. We\u0027re supposed to put them"},{"Start":"03:21.380 ","End":"03:27.240","Text":"in absolute value even though I can see that t is positive."},{"Start":"03:27.920 ","End":"03:31.670","Text":"Then plus a constant but I"},{"Start":"03:31.670 ","End":"03:35.940","Text":"won\u0027t bother with the constant because we\u0027re going to do a definite integral."},{"Start":"03:36.140 ","End":"03:44.090","Text":"Then we have to substitute back instead of t e to the power of x, here it is."},{"Start":"03:44.090 ","End":"03:50.310","Text":"This is equal to natural log of e^x that\u0027s positive,"},{"Start":"03:50.310 ","End":"03:55.265","Text":"so I dispense with the absolute value minus natural log."},{"Start":"03:55.265 ","End":"03:56.690","Text":"Here 1 plus e^x,"},{"Start":"03:56.690 ","End":"04:01.570","Text":"again is positive so I will dispense with the absolute value."},{"Start":"04:01.570 ","End":"04:05.955","Text":"Finally, natural log of e to the power of"},{"Start":"04:05.955 ","End":"04:09.680","Text":"is just x itself because these functions are reverse of each other,"},{"Start":"04:09.680 ","End":"04:12.550","Text":"exponent and natural log are the reverse of each other."},{"Start":"04:12.550 ","End":"04:22.330","Text":"We\u0027re back with x minus natural log of 1 plus e^x plus a constant for those who insist."},{"Start":"04:22.330 ","End":"04:28.220","Text":"Let me just copy that and then we\u0027ll be able to scroll back up."},{"Start":"04:28.220 ","End":"04:31.070","Text":"What we want is this thing."},{"Start":"04:31.070 ","End":"04:34.145","Text":"This is already the integral."},{"Start":"04:34.145 ","End":"04:37.520","Text":"What we want to do is evaluate it."},{"Start":"04:37.520 ","End":"04:40.010","Text":"There is a limit also, yeah."},{"Start":"04:40.010 ","End":"04:45.380","Text":"Evaluate it between 0 and b."},{"Start":"04:45.380 ","End":"04:47.870","Text":"Did I write 0? I meant infinity."},{"Start":"04:47.870 ","End":"04:50.060","Text":"Let\u0027s see what we have here."},{"Start":"04:50.060 ","End":"04:53.205","Text":"First of all, we substitute b."},{"Start":"04:53.205 ","End":"05:04.265","Text":"We get b minus natural log of 1 plus e to the power of b,"},{"Start":"05:04.265 ","End":"05:08.970","Text":"and then minus the same thing with 0."},{"Start":"05:08.970 ","End":"05:16.045","Text":"So it\u0027s 0 minus natural log of 1 plus e^0."},{"Start":"05:16.045 ","End":"05:21.950","Text":"Then this whole thing limit as b goes to infinity,"},{"Start":"05:21.950 ","End":"05:24.335","Text":"better put an extra brackets here."},{"Start":"05:24.335 ","End":"05:28.220","Text":"Let\u0027s see what we have now, 0 is 0,"},{"Start":"05:28.220 ","End":"05:32.370","Text":"e^0 is 1,1 plus 1 is 2,"},{"Start":"05:32.370 ","End":"05:35.690","Text":"natural log of 2 and there is a minus and a minus."},{"Start":"05:35.690 ","End":"05:39.740","Text":"This bit is plus natural log of 2."},{"Start":"05:39.740 ","End":"05:42.005","Text":"What\u0027s the first bit?"},{"Start":"05:42.005 ","End":"05:49.145","Text":"The first bit, it\u0027s b minus the natural log of 1 plus e^b."},{"Start":"05:49.145 ","End":"05:52.265","Text":"How are we going to figure out the limit of that?"},{"Start":"05:52.265 ","End":"05:54.665","Text":"Why is there a difficulty?"},{"Start":"05:54.665 ","End":"06:01.415","Text":"Because I can already see it\u0027s a case of infinity minus infinity so we want the limit."},{"Start":"06:01.415 ","End":"06:03.905","Text":"Actually, we can just take the limit of this part."},{"Start":"06:03.905 ","End":"06:06.560","Text":"The natural log of 2 makes no difference."},{"Start":"06:06.560 ","End":"06:13.170","Text":"It\u0027s the limit as b goes to infinity of this."},{"Start":"06:13.170 ","End":"06:16.760","Text":"The question is, as b goes to infinity,"},{"Start":"06:16.760 ","End":"06:20.140","Text":"that\u0027s infinity and e_b goes to infinity."},{"Start":"06:20.140 ","End":"06:23.510","Text":"It\u0027s not clear what this limit is,"},{"Start":"06:23.510 ","End":"06:26.540","Text":"but we\u0027re going to have to use some tricks here."},{"Start":"06:26.540 ","End":"06:29.930","Text":"What I\u0027d like to do is compute just this bit,"},{"Start":"06:29.930 ","End":"06:33.740","Text":"the natural log of 1 plus e^b at the side,"},{"Start":"06:33.740 ","End":"06:36.060","Text":"I\u0027ll call it asterisk."},{"Start":"06:37.130 ","End":"06:40.785","Text":"Then afterwards I\u0027ll return to this step."},{"Start":"06:40.785 ","End":"06:45.830","Text":"Natural log of 1 plus e to the power of"},{"Start":"06:45.830 ","End":"06:50.570","Text":"b. I\u0027m going to take e^b outside the brackets so it\u0027s"},{"Start":"06:50.570 ","End":"06:54.949","Text":"a natural log of e^b times"},{"Start":"06:54.949 ","End":"07:02.220","Text":"1 plus 1 over e to the power of b."},{"Start":"07:02.220 ","End":"07:04.470","Text":"Better put a bracket here."},{"Start":"07:04.470 ","End":"07:08.930","Text":"Since the log of a product is the sum of the logs,"},{"Start":"07:08.930 ","End":"07:18.620","Text":"this is natural log of e^b plus natural log of 1 plus 1 over e^b."},{"Start":"07:18.620 ","End":"07:22.340","Text":"Now, this is going to simplify nicely because"},{"Start":"07:22.340 ","End":"07:26.225","Text":"natural log of e to the power of is just b itself."},{"Start":"07:26.225 ","End":"07:28.340","Text":"I mean these 2 functions are opposite of each other."},{"Start":"07:28.340 ","End":"07:30.020","Text":"If you take the exponent and the logarithm,"},{"Start":"07:30.020 ","End":"07:31.910","Text":"you\u0027re back to the number itself."},{"Start":"07:31.910 ","End":"07:38.390","Text":"This finally equals just b plus natural log"},{"Start":"07:38.390 ","End":"07:45.010","Text":"of 1 plus, I\u0027ll write it as e^minus b."},{"Start":"07:45.740 ","End":"07:48.930","Text":"Now, that\u0027s the asterisk."},{"Start":"07:48.930 ","End":"07:51.815","Text":"If I put the asterisk back in,"},{"Start":"07:51.815 ","End":"07:56.299","Text":"then what I will get continuing here equals,"},{"Start":"07:56.299 ","End":"08:03.210","Text":"this is from here b minus this asterisk which is this."},{"Start":"08:03.210 ","End":"08:08.260","Text":"It\u0027s minus b minus this if I open the brackets,"},{"Start":"08:08.260 ","End":"08:12.995","Text":"plus natural log of 2 and all this,"},{"Start":"08:12.995 ","End":"08:16.435","Text":"the limit, well we can put the limit just of this bit."},{"Start":"08:16.435 ","End":"08:21.510","Text":"Limit as b goes to infinity."},{"Start":"08:21.510 ","End":"08:26.290","Text":"Now note that this b cancels with this b."},{"Start":"08:28.460 ","End":"08:32.085","Text":"Sorry, I wrote this 0, this is infinity."},{"Start":"08:32.085 ","End":"08:38.210","Text":"What we get basically is just minus the natural log of"},{"Start":"08:38.210 ","End":"08:44.870","Text":"1 plus e^minus infinity plus natural log of 2."},{"Start":"08:44.870 ","End":"08:49.520","Text":"Now we\u0027re getting very close to the end, because e^minus infinity,"},{"Start":"08:49.520 ","End":"08:52.220","Text":"this bit is just 0,"},{"Start":"08:52.220 ","End":"08:54.395","Text":"0 plus 1 is 1,"},{"Start":"08:54.395 ","End":"08:58.280","Text":"natural log of 1 is 0 minus 0 is still 0."},{"Start":"08:58.280 ","End":"08:59.990","Text":"All we\u0027re left with is this,"},{"Start":"08:59.990 ","End":"09:03.425","Text":"which is the natural log of 2,"},{"Start":"09:03.425 ","End":"09:09.720","Text":"and that is the answer. We are done."}],"ID":4583},{"Watched":false,"Name":"Exercise 14","Duration":"2m 15s","ChapterTopicVideoID":4575,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.024","Text":"In this exercise, we have to compute the following improper integral."},{"Start":"00:05.024 ","End":"00:08.025","Text":"It\u0027s improper because there\u0027s an infinity here,"},{"Start":"00:08.025 ","End":"00:10.830","Text":"it might converge or it might diverge."},{"Start":"00:10.830 ","End":"00:14.325","Text":"If it converges, then we should also evaluate it."},{"Start":"00:14.325 ","End":"00:17.910","Text":"What does it mean that there is an infinity here?"},{"Start":"00:17.910 ","End":"00:20.635","Text":"It means that we take the limit,"},{"Start":"00:20.635 ","End":"00:26.090","Text":"some lateral call it b goes to infinity of the same thing"},{"Start":"00:26.090 ","End":"00:34.100","Text":"from 0-b instead of infinity of cosine x dx."},{"Start":"00:34.100 ","End":"00:39.635","Text":"Now, this definite integral will be solved with the help of the indefinite integral,"},{"Start":"00:39.635 ","End":"00:45.065","Text":"the indefinite integral of cosine x is just sine x and we don\u0027t need a constant."},{"Start":"00:45.065 ","End":"00:51.810","Text":"What we have here is sine x taken from 0-b,"},{"Start":"00:51.810 ","End":"00:56.840","Text":"and then we take the limit as b goes to infinity."},{"Start":"00:56.840 ","End":"01:00.080","Text":"Now, this notation just means we substitute b,"},{"Start":"01:00.080 ","End":"01:02.755","Text":"we substitute 0 and we subtract."},{"Start":"01:02.755 ","End":"01:07.320","Text":"First I\u0027ll write down the limit as to a subtraction."},{"Start":"01:07.320 ","End":"01:10.440","Text":"I put in b, I get sine b."},{"Start":"01:10.440 ","End":"01:15.780","Text":"I put in 0, I get sine 0."},{"Start":"01:15.780 ","End":"01:17.520","Text":"Sine of 0 is 0,"},{"Start":"01:17.520 ","End":"01:23.705","Text":"so I\u0027m just left with the limit as b goes to infinity of the sine of b."},{"Start":"01:23.705 ","End":"01:25.890","Text":"Now what is this limit?"},{"Start":"01:25.890 ","End":"01:29.390","Text":"In fact this is one of those limits that does not exist,"},{"Start":"01:29.390 ","End":"01:31.730","Text":"it\u0027s not that it\u0027s infinity or minus infinity,"},{"Start":"01:31.730 ","End":"01:35.450","Text":"it doesn\u0027t exist and if I drew a very quick sketch you would see"},{"Start":"01:35.450 ","End":"01:40.325","Text":"if we have this as, say the y-axis and the x-axis,"},{"Start":"01:40.325 ","End":"01:43.490","Text":"the sine always stays between 1 and minus 1,"},{"Start":"01:43.490 ","End":"01:45.670","Text":"but it keeps oscillating."},{"Start":"01:45.670 ","End":"01:48.600","Text":"Even if we get so on and so on and so on,"},{"Start":"01:48.600 ","End":"01:52.220","Text":"near the hundreds of thousands or trillions of zillions,"},{"Start":"01:52.220 ","End":"01:56.770","Text":"it\u0027s still going to go up and down forever,"},{"Start":"01:56.770 ","End":"01:59.060","Text":"and it\u0027s never going to get close to something."},{"Start":"01:59.060 ","End":"02:02.790","Text":"It\u0027s always going to have values of 1 minus 1 and everything in between,"},{"Start":"02:02.790 ","End":"02:05.600","Text":"it\u0027s never getting close to 1 single thing."},{"Start":"02:05.600 ","End":"02:07.655","Text":"This does not exist."},{"Start":"02:07.655 ","End":"02:13.140","Text":"This simply diverges and we can\u0027t evaluate it,"},{"Start":"02:13.140 ","End":"02:14.669","Text":"that\u0027s it, diverges."},{"Start":"02:14.669 ","End":"02:16.690","Text":"I\u0027m Done."}],"ID":4584},{"Watched":false,"Name":"Exercise 15","Duration":"5m 3s","ChapterTopicVideoID":4576,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Here we\u0027re given an improper integral,"},{"Start":"00:02.280 ","End":"00:04.980","Text":"I know this because of the infinity here."},{"Start":"00:04.980 ","End":"00:07.680","Text":"It could converge or diverge,"},{"Start":"00:07.680 ","End":"00:09.089","Text":"you have to decide which,"},{"Start":"00:09.089 ","End":"00:10.260","Text":"and if it converges,"},{"Start":"00:10.260 ","End":"00:12.360","Text":"you have to also evaluate it."},{"Start":"00:12.360 ","End":"00:15.690","Text":"Well, let\u0027s see. What do we do with this infinity?"},{"Start":"00:15.690 ","End":"00:17.264","Text":"The answer is simple."},{"Start":"00:17.264 ","End":"00:18.990","Text":"Instead of taking an infinity,"},{"Start":"00:18.990 ","End":"00:22.230","Text":"we just take some variable that goes to infinity usually the"},{"Start":"00:22.230 ","End":"00:27.045","Text":"letter b and we let b go to infinity."},{"Start":"00:27.045 ","End":"00:28.620","Text":"The rest is the same,"},{"Start":"00:28.620 ","End":"00:34.205","Text":"0 x squared e to the minus x dx."},{"Start":"00:34.205 ","End":"00:39.885","Text":"We need to know what the indefinite integral of x squared e to the minus x is."},{"Start":"00:39.885 ","End":"00:41.595","Text":"I\u0027ll do that aside,"},{"Start":"00:41.595 ","End":"00:46.235","Text":"the integral of x squared e to the minus x dx."},{"Start":"00:46.235 ","End":"00:48.560","Text":"This is actually quite lengthy."},{"Start":"00:48.560 ","End":"00:53.510","Text":"It\u0027s integration by parts twice so I won\u0027t actually do it,"},{"Start":"00:53.510 ","End":"00:54.650","Text":"I\u0027ll just write down."},{"Start":"00:54.650 ","End":"00:57.575","Text":"If you want to try it at home you do it by parts,"},{"Start":"00:57.575 ","End":"01:01.015","Text":"and then you get an integral of x e to the minus x."},{"Start":"01:01.015 ","End":"01:04.835","Text":"You actually have to end up doing it by parts twice."},{"Start":"01:04.835 ","End":"01:05.990","Text":"Now I\u0027ll just let you know,"},{"Start":"01:05.990 ","End":"01:09.710","Text":"the answer is minus x squared,"},{"Start":"01:09.710 ","End":"01:13.000","Text":"minus 2x minus 2,"},{"Start":"01:13.000 ","End":"01:16.890","Text":"e to the minus x and if you want plus c,"},{"Start":"01:16.890 ","End":"01:20.400","Text":"but we won\u0027t need the c. If you want to,"},{"Start":"01:20.400 ","End":"01:23.360","Text":"just check it by differentiating this to see that you get"},{"Start":"01:23.360 ","End":"01:26.720","Text":"this and then you\u0027ll be more reassured or try doing it as an exercise."},{"Start":"01:26.720 ","End":"01:28.940","Text":"Anyway, I\u0027m coming back here."},{"Start":"01:28.940 ","End":"01:33.465","Text":"Limit as b goes to infinity."},{"Start":"01:33.465 ","End":"01:40.070","Text":"No more integrals, I just have some expression which is the minus x"},{"Start":"01:40.070 ","End":"01:47.570","Text":"squared minus 2x minus 2 times e to the minus x."},{"Start":"01:47.570 ","End":"01:51.275","Text":"We have to evaluate it between 0 and b,"},{"Start":"01:51.275 ","End":"01:54.665","Text":"which means I substitute this, substitute this, subtract."},{"Start":"01:54.665 ","End":"01:57.109","Text":"If I substitute b,"},{"Start":"01:57.109 ","End":"02:00.200","Text":"the limit b goes to infinity."},{"Start":"02:00.200 ","End":"02:02.240","Text":"If I substitute b here,"},{"Start":"02:02.240 ","End":"02:08.510","Text":"I get minus b squared minus 2b minus 2,"},{"Start":"02:08.510 ","End":"02:12.800","Text":"all this times e to the minus b."},{"Start":"02:12.800 ","End":"02:15.110","Text":"If I substitute 0,"},{"Start":"02:15.110 ","End":"02:21.810","Text":"e to the minus 0 is 1 so I\u0027m just left with minus 2,"},{"Start":"02:21.810 ","End":"02:24.690","Text":"but minus minus 2 is plus 2."},{"Start":"02:24.690 ","End":"02:28.100","Text":"Well, I\u0027ll just leave it as minus minus 2 for the moment."},{"Start":"02:28.100 ","End":"02:31.070","Text":"Now what I\u0027m left with is,"},{"Start":"02:31.070 ","End":"02:33.660","Text":"the constant is just the constant is plus 2,"},{"Start":"02:33.660 ","End":"02:43.500","Text":"but this limit becomes b goes to infinity of minus b squared minus"},{"Start":"02:43.500 ","End":"02:48.460","Text":"2b minus 2 over e to the power of"},{"Start":"02:48.460 ","End":"02:50.410","Text":"b. I prefer it to have"},{"Start":"02:50.410 ","End":"02:53.905","Text":"a quotient because I\u0027m going to use L\u0027Hopital\u0027s rule as you\u0027ll see."},{"Start":"02:53.905 ","End":"02:58.730","Text":"This e to the minus b goes into the denominator as e to the power of b,"},{"Start":"02:58.730 ","End":"03:03.820","Text":"and then the plus 2, which can be inside or outside the limit, it doesn\u0027t matter."},{"Start":"03:03.820 ","End":"03:06.250","Text":"Now the question is, what is this limit?"},{"Start":"03:06.250 ","End":"03:09.130","Text":"Again, I\u0027m going to do a side exercise."},{"Start":"03:09.130 ","End":"03:15.025","Text":"The limit as b goes to infinity of minus b squared"},{"Start":"03:15.025 ","End":"03:21.330","Text":"minus 2b minus 2 over e to the b is equal to,"},{"Start":"03:21.330 ","End":"03:25.370","Text":"now this is a polynomial with negative coefficient of b"},{"Start":"03:25.370 ","End":"03:29.980","Text":"squared and this is the 1 that determines what the limit is,"},{"Start":"03:29.980 ","End":"03:33.080","Text":"whether it\u0027s infinity or minus infinity or a number,"},{"Start":"03:33.080 ","End":"03:35.060","Text":"but because it\u0027s negative here,"},{"Start":"03:35.060 ","End":"03:37.625","Text":"this thing actually goes to minus infinity."},{"Start":"03:37.625 ","End":"03:41.330","Text":"You can also take b squared outside the brackets and"},{"Start":"03:41.330 ","End":"03:44.870","Text":"you\u0027ll get minus b squared times something that goes to 1."},{"Start":"03:44.870 ","End":"03:47.195","Text":"In any event, this is minus infinity,"},{"Start":"03:47.195 ","End":"03:51.830","Text":"and e to the b is infinity when b goes to infinity."},{"Start":"03:51.830 ","End":"03:53.630","Text":"That\u0027s what we get, and I\u0027m going to say equals,"},{"Start":"03:53.630 ","End":"03:55.235","Text":"which is symbolically equal to."},{"Start":"03:55.235 ","End":"04:02.585","Text":"It means that we can use L\u0027Hopital\u0027s rule and get the limit as b goes to infinity."},{"Start":"04:02.585 ","End":"04:04.100","Text":"Just write his name,"},{"Start":"04:04.100 ","End":"04:06.340","Text":"it doesn\u0027t often get written."},{"Start":"04:06.340 ","End":"04:10.715","Text":"L\u0027Hopital\u0027s rule for infinity over infinity."},{"Start":"04:10.715 ","End":"04:12.200","Text":"We differentiate the top,"},{"Start":"04:12.200 ","End":"04:14.990","Text":"it\u0027s minus 2b minus 2,"},{"Start":"04:14.990 ","End":"04:17.010","Text":"differentiate the bottom e to the b,"},{"Start":"04:17.010 ","End":"04:18.080","Text":"we\u0027ve got a new integral."},{"Start":"04:18.080 ","End":"04:20.855","Text":"Does this converge to something?"},{"Start":"04:20.855 ","End":"04:24.620","Text":"Again, it\u0027s L\u0027Hopital because we have minus infinity over infinity,"},{"Start":"04:24.620 ","End":"04:32.195","Text":"we\u0027ll keep going and then we\u0027ll get the limit as b goes to infinity minus 2b minus 2."},{"Start":"04:32.195 ","End":"04:36.270","Text":"It\u0027s just minus 2 over e to the b."},{"Start":"04:36.270 ","End":"04:38.360","Text":"Here we had minus infinity over infinity."},{"Start":"04:38.360 ","End":"04:44.665","Text":"Here we now have minus 2 over infinity, which is 0."},{"Start":"04:44.665 ","End":"04:47.840","Text":"If all this goes to 0 and I\u0027ll just"},{"Start":"04:47.840 ","End":"04:51.395","Text":"circle it because that was what we did at the side there,"},{"Start":"04:51.395 ","End":"04:56.255","Text":"then our answer will equal just 2."},{"Start":"04:56.255 ","End":"05:00.140","Text":"Notice this is 0 plus 2 is 2 and we are done."},{"Start":"05:00.140 ","End":"05:03.960","Text":"I just didn\u0027t do the integration by parts."}],"ID":4585},{"Watched":false,"Name":"Exercise 16","Duration":"4m 10s","ChapterTopicVideoID":4577,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"Here I have an improper integral;"},{"Start":"00:02.190 ","End":"00:04.650","Text":"I can tell by the infinity."},{"Start":"00:04.650 ","End":"00:09.960","Text":"We have to determine if it converges or diverges and if it converges,"},{"Start":"00:09.960 ","End":"00:12.375","Text":"we\u0027ve got to evaluate it. Let\u0027s see."},{"Start":"00:12.375 ","End":"00:16.020","Text":"Copy it over here quickly, 1 to infinity,"},{"Start":"00:16.020 ","End":"00:22.635","Text":"x over 1 plus x squared, squared dx."},{"Start":"00:22.635 ","End":"00:26.430","Text":"First, I\u0027m going to interpret what the infinity means."},{"Start":"00:26.430 ","End":"00:30.300","Text":"The infinity means something that tends to infinity,"},{"Start":"00:30.300 ","End":"00:32.505","Text":"so I\u0027ll use some letter say b,"},{"Start":"00:32.505 ","End":"00:36.890","Text":"and let b tend to infinity and the rest of it,"},{"Start":"00:36.890 ","End":"00:38.375","Text":"I copy the same,"},{"Start":"00:38.375 ","End":"00:43.340","Text":"x over 1 plus x squared, squared dx."},{"Start":"00:43.340 ","End":"00:47.300","Text":"This is a regular definite integral although b"},{"Start":"00:47.300 ","End":"00:51.155","Text":"is a parameter and to do the definite integral,"},{"Start":"00:51.155 ","End":"00:54.665","Text":"it\u0027s useful to have the indefinite integral, the anti-derivative."},{"Start":"00:54.665 ","End":"00:56.390","Text":"I\u0027ll do that at the side here,"},{"Start":"00:56.390 ","End":"01:00.665","Text":"so let us see what is the integral of"},{"Start":"01:00.665 ","End":"01:06.775","Text":"x over 1 plus x squared, squared."},{"Start":"01:06.775 ","End":"01:11.000","Text":"To me, it looks like a case for substitution because I have"},{"Start":"01:11.000 ","End":"01:15.350","Text":"1 plus x squared and I also have its derivative, well, almost."},{"Start":"01:15.350 ","End":"01:18.305","Text":"Its derivative would be 2x."},{"Start":"01:18.305 ","End":"01:24.885","Text":"How about I put a 2 here and 1/2 here?"},{"Start":"01:24.885 ","End":"01:27.460","Text":"That shouldn\u0027t change anything, should it?"},{"Start":"01:27.460 ","End":"01:32.100","Text":"Now I\u0027ll take, let\u0027s see what letter t usually,"},{"Start":"01:32.100 ","End":"01:34.965","Text":"t will be 1 plus x squared,"},{"Start":"01:34.965 ","End":"01:40.765","Text":"dt will be the derivative of this 2x dx."},{"Start":"01:40.765 ","End":"01:45.845","Text":"After we substitute, what we\u0027re going to get is 1/2."},{"Start":"01:45.845 ","End":"01:50.760","Text":"Now the 2x dx is just dt."},{"Start":"01:50.760 ","End":"01:53.790","Text":"1 plus x squared is t,"},{"Start":"01:53.790 ","End":"01:56.460","Text":"so this makes this t squared."},{"Start":"01:56.460 ","End":"02:04.060","Text":"Now, 1 over t squared is like t to the minus 2 and so in the case of exponents,"},{"Start":"02:04.060 ","End":"02:07.765","Text":"we just increased the exponent by 1 and divide by it."},{"Start":"02:07.765 ","End":"02:17.560","Text":"We get 1/2 times t to the minus 1 over minus 1 plus the constant."},{"Start":"02:17.560 ","End":"02:20.800","Text":"The minus in the denominator could be in the numerator,"},{"Start":"02:20.800 ","End":"02:23.270","Text":"it\u0027s just minus 1/2."},{"Start":"02:24.500 ","End":"02:28.330","Text":"But the t also could go into the denominator would be 1 over 2."},{"Start":"02:28.330 ","End":"02:32.520","Text":"Basically, we get minus 1 over 2t."},{"Start":"02:32.520 ","End":"02:35.990","Text":"Again, the t minus 1 is 1 over t. The minus"},{"Start":"02:35.990 ","End":"02:40.750","Text":"1 becomes a minus here and this 1/2 is the 2 in the denominator."},{"Start":"02:40.750 ","End":"02:47.630","Text":"Now we have to switch back from t to x. I\u0027ll just do it right"},{"Start":"02:47.630 ","End":"02:54.290","Text":"here and say that what we get is still limit as b goes to infinity."},{"Start":"02:54.290 ","End":"03:00.305","Text":"But now we have the integral and the integral is minus 1 over 2."},{"Start":"03:00.305 ","End":"03:10.430","Text":"Now instead of t, I\u0027m going to replace it by 1 plus x squared so twice 1 plus x squared."},{"Start":"03:10.430 ","End":"03:13.390","Text":"Oh, I wrote by mistake."},{"Start":"03:13.390 ","End":"03:19.095","Text":"Fix that, and then we want it from 1 to b,"},{"Start":"03:19.095 ","End":"03:22.290","Text":"so we just substitute the b then the 1, and subtract."},{"Start":"03:22.290 ","End":"03:23.520","Text":"What we get is,"},{"Start":"03:23.520 ","End":"03:25.839","Text":"we still have a limit."},{"Start":"03:25.880 ","End":"03:28.230","Text":"When I put b in,"},{"Start":"03:28.230 ","End":"03:33.855","Text":"I get minus 1 over twice 1 plus b squared."},{"Start":"03:33.855 ","End":"03:36.310","Text":"When I put 1 in,"},{"Start":"03:36.530 ","End":"03:41.735","Text":"close the bracket, I get minus minus, which is plus."},{"Start":"03:41.735 ","End":"03:45.470","Text":"If I put 1 in here, 1 squared is 1 plus 1 is 2, 2 times 2 is 4,"},{"Start":"03:45.470 ","End":"03:50.705","Text":"so plus 1/4 and b goes to infinity."},{"Start":"03:50.705 ","End":"03:53.195","Text":"Now if b goes to infinity,"},{"Start":"03:53.195 ","End":"03:56.075","Text":"then this b squared goes to infinity;"},{"Start":"03:56.075 ","End":"04:00.925","Text":"the whole denominator goes to infinity so we\u0027re left with 0 here."},{"Start":"04:00.925 ","End":"04:05.535","Text":"All we get is 0 plus a 1/4,"},{"Start":"04:05.535 ","End":"04:10.960","Text":"which is 1/4 and that is the answer."}],"ID":4586},{"Watched":false,"Name":"Exercise 17","Duration":"5m 8s","ChapterTopicVideoID":4578,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"Here we have an improper integral because there\u0027s an infinity here."},{"Start":"00:04.290 ","End":"00:08.550","Text":"If so, then it either converges or diverges."},{"Start":"00:08.550 ","End":"00:10.065","Text":"That\u0027s what we have to find out."},{"Start":"00:10.065 ","End":"00:13.560","Text":"If it converges, we also got to evaluate it."},{"Start":"00:13.560 ","End":"00:18.525","Text":"The first thing to do is to interpret the infinity."},{"Start":"00:18.525 ","End":"00:21.060","Text":"What we do is instead of the infinity,"},{"Start":"00:21.060 ","End":"00:23.595","Text":"we put some letter, usually b,"},{"Start":"00:23.595 ","End":"00:30.690","Text":"and take the limit as b goes to infinity and the rest of it is the same."},{"Start":"00:30.690 ","End":"00:39.285","Text":"1 here, 1 over 1 plus x times square root of x dx."},{"Start":"00:39.285 ","End":"00:42.934","Text":"Now I have a regular definite integral."},{"Start":"00:42.934 ","End":"00:44.680","Text":"But to compute a definite integral,"},{"Start":"00:44.680 ","End":"00:48.155","Text":"it helps to have the indefinite integral, the anti-derivative."},{"Start":"00:48.155 ","End":"00:51.380","Text":"Why don\u0027t I do that as a side exercise over here?"},{"Start":"00:51.380 ","End":"00:55.310","Text":"Let\u0027s see if we can figure out what is the integral of"},{"Start":"00:55.310 ","End":"01:01.765","Text":"1 over 1 plus x times square root of x dx."},{"Start":"01:01.765 ","End":"01:04.880","Text":"I would recommend a substitution."},{"Start":"01:04.880 ","End":"01:09.600","Text":"Let\u0027s try t equals the square root of x."},{"Start":"01:09.600 ","End":"01:14.910","Text":"Then dt is 1 over twice square root of x."},{"Start":"01:14.910 ","End":"01:19.870","Text":"This is immediate, you\u0027re supposed to remember stuff like this, dx."},{"Start":"01:19.940 ","End":"01:22.835","Text":"Let\u0027s see what we get here."},{"Start":"01:22.835 ","End":"01:27.650","Text":"We also might want to do it the other way around to say what dx is in terms of dt,"},{"Start":"01:27.650 ","End":"01:32.135","Text":"well, dx is equal to 2 square root of x dt."},{"Start":"01:32.135 ","End":"01:34.640","Text":"Armed with all this,"},{"Start":"01:34.640 ","End":"01:38.810","Text":"we now get the integral of 1 over."},{"Start":"01:38.810 ","End":"01:41.775","Text":"Now, let\u0027s see."},{"Start":"01:41.775 ","End":"01:46.650","Text":"1 is 1, x would be t squared,"},{"Start":"01:46.650 ","End":"01:53.805","Text":"if we just reverse this x is t squared so t squared."},{"Start":"01:53.805 ","End":"01:58.640","Text":"I see that we have the square root of x here,"},{"Start":"01:58.640 ","End":"02:00.595","Text":"which I could write as t,"},{"Start":"02:00.595 ","End":"02:10.295","Text":"but I don\u0027t actually need to because I see that my dx is twice the square root of x dt."},{"Start":"02:10.295 ","End":"02:14.880","Text":"Therefore, this cancels with this"},{"Start":"02:14.880 ","End":"02:18.600","Text":"and so I don\u0027t have x anymore and that\u0027s good."},{"Start":"02:19.720 ","End":"02:30.405","Text":"The 2 can come out and I\u0027m left with twice the integral of 1 over 1 plus t squared dt."},{"Start":"02:30.405 ","End":"02:33.725","Text":"This is one of those immediate integrals."},{"Start":"02:33.725 ","End":"02:36.905","Text":"The integral of this is the arctangent."},{"Start":"02:36.905 ","End":"02:42.510","Text":"I have twice arctangent of t."},{"Start":"02:42.510 ","End":"02:46.935","Text":"But we didn\u0027t have t originally,"},{"Start":"02:46.935 ","End":"02:48.975","Text":"t is the square root of x,"},{"Start":"02:48.975 ","End":"02:51.600","Text":"and so I can just say that this is,"},{"Start":"02:51.600 ","End":"02:52.920","Text":"well, let\u0027s write it down here."},{"Start":"02:52.920 ","End":"02:54.330","Text":"This is square root of x."},{"Start":"02:54.330 ","End":"02:55.680","Text":"Let\u0027s continue."},{"Start":"02:55.680 ","End":"03:04.820","Text":"This is equal to the limit as b goes to infinity of twice"},{"Start":"03:04.820 ","End":"03:12.040","Text":"the arctangent of the square root of x."},{"Start":"03:12.980 ","End":"03:18.650","Text":"This from 1 to b,"},{"Start":"03:18.650 ","End":"03:20.810","Text":"which means I substitute each of these"},{"Start":"03:20.810 ","End":"03:23.550","Text":"and subtract the bottom from the top."},{"Start":"03:23.740 ","End":"03:27.860","Text":"The 2 will come out in front of the limit,"},{"Start":"03:27.860 ","End":"03:33.330","Text":"so I\u0027ve got twice the limit as b goes to infinity"},{"Start":"03:33.330 ","End":"03:35.445","Text":"and I remember its 2 is in front."},{"Start":"03:35.445 ","End":"03:45.105","Text":"I have the arctangent of square root of b minus the arctangent of the square root of 1."},{"Start":"03:45.105 ","End":"03:47.895","Text":"It\u0027s arctangent of 1."},{"Start":"03:47.895 ","End":"03:53.140","Text":"All this, I have to take the limit as b goes to infinity."},{"Start":"03:53.140 ","End":"03:55.685","Text":"If b goes to infinity,"},{"Start":"03:55.685 ","End":"03:58.715","Text":"square root of b also goes to infinity."},{"Start":"03:58.715 ","End":"04:04.155","Text":"I end up getting twice arctangent"},{"Start":"04:04.155 ","End":"04:10.800","Text":"of infinity minus arctangent of 1."},{"Start":"04:10.800 ","End":"04:13.169","Text":"Both of these have a value."},{"Start":"04:13.169 ","End":"04:16.475","Text":"The arctangent of infinity is,"},{"Start":"04:16.475 ","End":"04:18.350","Text":"if I think of it in regular trigonometry,"},{"Start":"04:18.350 ","End":"04:23.270","Text":"the angle where the tangent becomes infinity is 90 degrees."},{"Start":"04:23.270 ","End":"04:25.280","Text":"Maybe I\u0027ll make a little note just because"},{"Start":"04:25.280 ","End":"04:29.720","Text":"I still think in degrees that this would be 90 degrees,"},{"Start":"04:29.720 ","End":"04:31.760","Text":"the angle whose tangent is infinity"},{"Start":"04:31.760 ","End":"04:35.765","Text":"and the angle whose tangent is 1 is 45 degrees that I know."},{"Start":"04:35.765 ","End":"04:42.545","Text":"But use your calculators and what we get in our case of course is in radians."},{"Start":"04:42.545 ","End":"04:50.090","Text":"90 degrees is just Pi over 2 and 45 degrees is Pi over 4."},{"Start":"04:50.090 ","End":"04:52.280","Text":"Twice of all of this."},{"Start":"04:52.280 ","End":"04:57.700","Text":"Let\u0027s see, Pi over 2 minus Pi over 4 is Pi over 4."},{"Start":"04:57.700 ","End":"05:00.845","Text":"Pi over 4 times 2 is Pi over 2."},{"Start":"05:00.845 ","End":"05:02.730","Text":"I make the final answer,"},{"Start":"05:02.730 ","End":"05:06.580","Text":"Pi over 2 and I\u0027ll put it in a little box."},{"Start":"05:06.580 ","End":"05:09.610","Text":"Okay, we are done."}],"ID":4587},{"Watched":false,"Name":"Exercise 18","Duration":"5m 38s","ChapterTopicVideoID":4579,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"Here we have an improper integral."},{"Start":"00:02.010 ","End":"00:04.605","Text":"I know this from the minus infinity."},{"Start":"00:04.605 ","End":"00:08.520","Text":"What we have to do is decide whether it converges or diverges."},{"Start":"00:08.520 ","End":"00:12.030","Text":"If it converges, we can actually compute the value."},{"Start":"00:12.030 ","End":"00:15.675","Text":"First, we want to interpret this minus infinity."},{"Start":"00:15.675 ","End":"00:17.520","Text":"I\u0027ll just copy the exercise again."},{"Start":"00:17.520 ","End":"00:20.200","Text":"Minus infinity to 0,"},{"Start":"00:20.570 ","End":"00:25.330","Text":"e^x/3 minus 2e^x dx."},{"Start":"00:27.380 ","End":"00:33.495","Text":"In this case, the minus infinity is replaced by a letter, let\u0027s call it a,"},{"Start":"00:33.495 ","End":"00:39.540","Text":"and then take the limit as a goes to minus infinity."},{"Start":"00:39.540 ","End":"00:45.255","Text":"That\u0027s what we do instead of the minus infinity without something 10 to minus infinity."},{"Start":"00:45.255 ","End":"00:47.565","Text":"Then the rest of it is the same,"},{"Start":"00:47.565 ","End":"00:55.075","Text":"0 here the e^x over 3 minus 2e^x dx."},{"Start":"00:55.075 ","End":"00:58.100","Text":"Now in order to do a definite integral,"},{"Start":"00:58.100 ","End":"00:59.240","Text":"which is what we have here,"},{"Start":"00:59.240 ","End":"01:01.250","Text":"it\u0027s good to know the indefinite integral,"},{"Start":"01:01.250 ","End":"01:03.965","Text":"that\u0027s the antiderivative or primitive."},{"Start":"01:03.965 ","End":"01:07.935","Text":"I\u0027ll do that at the side, so change color,"},{"Start":"01:07.935 ","End":"01:13.010","Text":"and let\u0027s see if we can compute the integral of"},{"Start":"01:13.010 ","End":"01:22.685","Text":"e^x/3 minus 2 e^x dx."},{"Start":"01:22.685 ","End":"01:27.320","Text":"This looks to me very much like an integration by substitution,"},{"Start":"01:27.320 ","End":"01:34.895","Text":"and the substitution that seems most logical to try is to let t equals e^x."},{"Start":"01:34.895 ","End":"01:37.820","Text":"Then what would d t equal the derivative of this,"},{"Start":"01:37.820 ","End":"01:40.140","Text":"which is itself dx."},{"Start":"01:40.140 ","End":"01:45.410","Text":"This looks very good to me because I already have e^x dx there."},{"Start":"01:45.410 ","End":"01:51.805","Text":"If I substitute, what I\u0027ll get is the integral of e^x dx is just dt."},{"Start":"01:51.805 ","End":"01:56.085","Text":"On the denominator, I have 3 minus 2,"},{"Start":"01:56.085 ","End":"02:01.820","Text":"and e^x is t. If I just modify it a little bit,"},{"Start":"02:01.820 ","End":"02:05.580","Text":"it looks like one of those logarithm ones."},{"Start":"02:07.490 ","End":"02:10.170","Text":"Because it\u0027s not t,"},{"Start":"02:10.170 ","End":"02:13.460","Text":"but it\u0027s a linear function of t,"},{"Start":"02:13.460 ","End":"02:19.190","Text":"I can say that this is equal to the natural log at"},{"Start":"02:19.190 ","End":"02:25.850","Text":"first of 3 minus 2t, actually supposed to put it in absolute value of 3 minus 2t."},{"Start":"02:25.850 ","End":"02:29.600","Text":"But because it\u0027s a linear function of t not t,"},{"Start":"02:29.600 ","End":"02:31.970","Text":"we divide by the coefficient of t,"},{"Start":"02:31.970 ","End":"02:33.875","Text":"and this is the inner derivative,"},{"Start":"02:33.875 ","End":"02:37.055","Text":"but we can only do this with integration when it\u0027s linear."},{"Start":"02:37.055 ","End":"02:41.030","Text":"Minus 2, I have to divide by it over minus 2,"},{"Start":"02:41.030 ","End":"02:46.865","Text":"let me just put minus 1/5 in front plus a constant, and finally,"},{"Start":"02:46.865 ","End":"02:54.260","Text":"to substitute back from t 2 e^x. I get minus 1/5"},{"Start":"02:54.260 ","End":"03:00.720","Text":"natural log of 3 minus 2 e^x"},{"Start":"03:01.580 ","End":"03:06.060","Text":"plus c. That\u0027s the indefinite integral."},{"Start":"03:06.060 ","End":"03:08.145","Text":"Now I\u0027m going to go back here,"},{"Start":"03:08.145 ","End":"03:12.170","Text":"and just continue but substituting this."},{"Start":"03:12.170 ","End":"03:18.350","Text":"What I have is I still have a limit as a goes to minus infinity,"},{"Start":"03:18.350 ","End":"03:23.625","Text":"but this definite integral becomes minus 1/2."},{"Start":"03:23.625 ","End":"03:30.220","Text":"Let me put the minus 1/2 in front of the limit equals minus 1/2."},{"Start":"03:30.220 ","End":"03:34.175","Text":"The limit as a goes to infinity of"},{"Start":"03:34.175 ","End":"03:43.265","Text":"natural logarithm of absolute value of 3 minus 2 e^x,"},{"Start":"03:43.265 ","End":"03:48.710","Text":"and all this between a and 0."},{"Start":"03:48.710 ","End":"03:55.010","Text":"Next thing to do is just to substitute a and to substitute 0,"},{"Start":"03:55.010 ","End":"03:57.650","Text":"and subtract this one from this one."},{"Start":"03:57.650 ","End":"04:00.500","Text":"If I substitute 0,"},{"Start":"04:00.500 ","End":"04:05.505","Text":"e to the 0 is 1,"},{"Start":"04:05.505 ","End":"04:09.060","Text":"2,1 is 2, 3 minus 2 is 1,"},{"Start":"04:09.060 ","End":"04:11.130","Text":"so the first bit is 1."},{"Start":"04:11.130 ","End":"04:21.810","Text":"Then minus natural log of 3 minus 2 e^a,"},{"Start":"04:21.830 ","End":"04:26.350","Text":"and this is equal to minus 1/5."},{"Start":"04:26.350 ","End":"04:29.120","Text":"Now I can take the limit already,"},{"Start":"04:29.120 ","End":"04:31.440","Text":"so 1 just stays 1,"},{"Start":"04:31.440 ","End":"04:32.810","Text":"and if I apply the limit here,"},{"Start":"04:32.810 ","End":"04:40.250","Text":"I\u0027ll write it as the natural logarithm of absolute value of 3 minus 2 e^minus infinity,"},{"Start":"04:40.250 ","End":"04:46.170","Text":"because it\u0027s an expression which is defined actually, absolute values there."},{"Start":"04:48.850 ","End":"04:54.435","Text":"Let\u0027s see, e^minus infinity, this bit is 0."},{"Start":"04:54.435 ","End":"04:57.335","Text":"If this is 0, twice 0 is 0,"},{"Start":"04:57.335 ","End":"04:59.900","Text":"3 minus 0 is 3."},{"Start":"04:59.900 ","End":"05:04.095","Text":"Natural log of absolute value 3 is just natural log of 3."},{"Start":"05:04.095 ","End":"05:06.630","Text":"What I get is minus 1/2,"},{"Start":"05:06.630 ","End":"05:12.015","Text":"1 minus natural log of 3,"},{"Start":"05:12.015 ","End":"05:16.055","Text":"and that\u0027s about it."},{"Start":"05:16.055 ","End":"05:21.005","Text":"Natural log of 3 to bit over 1 because e is less than 3."},{"Start":"05:21.005 ","End":"05:23.975","Text":"I\u0027d actually like to write it without the minus,"},{"Start":"05:23.975 ","End":"05:27.785","Text":"let me write it as 1/5 of natural log of"},{"Start":"05:27.785 ","End":"05:31.790","Text":"3 minus 1 it\u0027ll come up positive because as I say,"},{"Start":"05:31.790 ","End":"05:33.455","Text":"natural log of e is 1,"},{"Start":"05:33.455 ","End":"05:34.910","Text":"log of 3 would be bigger than 1."},{"Start":"05:34.910 ","End":"05:38.850","Text":"That doesn\u0027t match anyway, this is the answer. We\u0027re done."}],"ID":4588},{"Watched":false,"Name":"Exercise 19","Duration":"3m 58s","ChapterTopicVideoID":4580,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"Here, we have to compute this improper integral."},{"Start":"00:03.225 ","End":"00:08.745","Text":"It\u0027s almost doubly improper because there\u0027s an infinity here and a minus infinity here."},{"Start":"00:08.745 ","End":"00:12.990","Text":"We\u0027ve got to decide whether it converges or diverges and if it converges,"},{"Start":"00:12.990 ","End":"00:14.700","Text":"we determine its value."},{"Start":"00:14.700 ","End":"00:16.620","Text":"I\u0027ve rewritten it here."},{"Start":"00:16.620 ","End":"00:19.965","Text":"The first thing is how to deal with these infinities."},{"Start":"00:19.965 ","End":"00:22.500","Text":"When both limits are infinite,"},{"Start":"00:22.500 ","End":"00:26.010","Text":"then we choose any point,"},{"Start":"00:26.010 ","End":"00:27.810","Text":"but make it convenient."},{"Start":"00:27.810 ","End":"00:32.340","Text":"I\u0027m going to choose 0 and split it up into 2 integrals,"},{"Start":"00:32.340 ","End":"00:38.180","Text":"from minus infinity to 0 of x cubed dx,"},{"Start":"00:38.180 ","End":"00:46.205","Text":"plus let\u0027s continue from 0 to infinity of x cubed dx."},{"Start":"00:46.205 ","End":"00:48.530","Text":"Then we work on each of these."},{"Start":"00:48.530 ","End":"00:51.695","Text":"Now, each of these is done in the similar fashion."},{"Start":"00:51.695 ","End":"00:54.485","Text":"Instead of the infinity plus or minus,"},{"Start":"00:54.485 ","End":"00:57.695","Text":"we take some letter that tends to that infinity."},{"Start":"00:57.695 ","End":"01:02.450","Text":"In this case, I\u0027ll use the letter a. I\u0027ll get the limit as"},{"Start":"01:02.450 ","End":"01:07.155","Text":"a goes to minus infinity, and here, I put a."},{"Start":"01:07.155 ","End":"01:09.110","Text":"Instead of the minus infinity and everything else"},{"Start":"01:09.110 ","End":"01:13.490","Text":"is the same, 0, x cubed dx."},{"Start":"01:13.490 ","End":"01:19.515","Text":"Here, I get the limit and this one I\u0027ll replace with b,"},{"Start":"01:19.515 ","End":"01:23.190","Text":"so I\u0027ll get b goes to infinity and here,"},{"Start":"01:23.190 ","End":"01:25.770","Text":"I\u0027ll have b instead of infinity and the rest,"},{"Start":"01:25.770 ","End":"01:29.900","Text":"the same, 0, x cubed dx."},{"Start":"01:29.900 ","End":"01:34.370","Text":"I\u0027m going to need the indefinite integral of x cubed,"},{"Start":"01:34.370 ","End":"01:37.130","Text":"but that\u0027s fairly obvious."},{"Start":"01:37.130 ","End":"01:47.145","Text":"The integral of x cubed dx as an indefinite integral is just raise the power by 1,"},{"Start":"01:47.145 ","End":"01:49.770","Text":"x^4 and divide by 4."},{"Start":"01:49.770 ","End":"01:52.050","Text":"With indefinite, you need a plus C,"},{"Start":"01:52.050 ","End":"01:54.210","Text":"but here, we don\u0027t need the plus C,"},{"Start":"01:54.210 ","End":"01:57.375","Text":"so it\u0027s one-quarter x to the fourth,"},{"Start":"01:57.375 ","End":"02:05.210","Text":"so this is equal to the limit as a goes to minus infinity."},{"Start":"02:05.210 ","End":"02:15.140","Text":"You know what? Let\u0027s take the quarter outside of x^4 between a and 0,"},{"Start":"02:15.140 ","End":"02:23.585","Text":"and the other one is one-quarter of the limit as b goes to plus infinity"},{"Start":"02:23.585 ","End":"02:32.760","Text":"also of x^4 between the limits of 0 and B."},{"Start":"02:32.760 ","End":"02:35.975","Text":"Let\u0027s see. Let\u0027s put the numbers in first."},{"Start":"02:35.975 ","End":"02:42.575","Text":"So we get one-quarter of the limit as a goes to minus infinity."},{"Start":"02:42.575 ","End":"02:45.170","Text":"What we get here if we put the 0 in,"},{"Start":"02:45.170 ","End":"02:50.980","Text":"we get 0^4 is 0 minus a^4,"},{"Start":"02:50.980 ","End":"02:52.715","Text":"and for the other one,"},{"Start":"02:52.715 ","End":"02:56.870","Text":"we get one-quarter of the limit."},{"Start":"02:56.870 ","End":"02:59.195","Text":"This time b goes to infinity."},{"Start":"02:59.195 ","End":"03:01.745","Text":"Again, just plug in b,"},{"Start":"03:01.745 ","End":"03:06.530","Text":"b^4 minus 0^4, and so on."},{"Start":"03:06.530 ","End":"03:12.055","Text":"What we get is if we let a go to minus infinity,"},{"Start":"03:12.055 ","End":"03:15.195","Text":"a^4 goes to plus infinity,"},{"Start":"03:15.195 ","End":"03:19.250","Text":"and so end up getting minus infinity."},{"Start":"03:19.250 ","End":"03:21.290","Text":"The quarter doesn\u0027t make any difference."},{"Start":"03:21.290 ","End":"03:23.810","Text":"So here we get minus infinity."},{"Start":"03:23.810 ","End":"03:26.060","Text":"What do we get here?"},{"Start":"03:26.060 ","End":"03:27.440","Text":"If b goes to infinity,"},{"Start":"03:27.440 ","End":"03:30.145","Text":"then b^4 is plus infinity,"},{"Start":"03:30.145 ","End":"03:32.910","Text":"and one-quarter of infinity is plus infinity,"},{"Start":"03:32.910 ","End":"03:36.620","Text":"so we get minus infinity plus infinity."},{"Start":"03:36.620 ","End":"03:39.125","Text":"When we get one of these forms,"},{"Start":"03:39.125 ","End":"03:42.305","Text":"these indeterminate undefined forms,"},{"Start":"03:42.305 ","End":"03:48.020","Text":"this implies that the original integral diverges."},{"Start":"03:48.020 ","End":"03:51.020","Text":"So this one does not converge,"},{"Start":"03:51.020 ","End":"03:56.930","Text":"it diverges, and we can\u0027t determine its value."},{"Start":"03:56.930 ","End":"03:59.400","Text":"Okay. That\u0027s it. We\u0027re done."}],"ID":4589},{"Watched":false,"Name":"Exercise 20","Duration":"4m 56s","ChapterTopicVideoID":4581,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"Here again, we have an improper integral because of the infinity."},{"Start":"00:04.410 ","End":"00:06.000","Text":"Well, it\u0027s doubly improper."},{"Start":"00:06.000 ","End":"00:07.995","Text":"We have infinity and minus infinity."},{"Start":"00:07.995 ","End":"00:11.730","Text":"We have to first decide whether it converges or diverges."},{"Start":"00:11.730 ","End":"00:14.835","Text":"If it converges, we also can determine its value."},{"Start":"00:14.835 ","End":"00:18.000","Text":"Now, how do we deal with infinity,"},{"Start":"00:18.000 ","End":"00:20.205","Text":"both at the top and at the bottom?"},{"Start":"00:20.205 ","End":"00:22.305","Text":"We break it up into 2."},{"Start":"00:22.305 ","End":"00:29.250","Text":"We say that this is equal to the integral of something to infinity,"},{"Start":"00:29.250 ","End":"00:36.525","Text":"plus the integral from minus infinity to that something,"},{"Start":"00:36.525 ","End":"00:41.060","Text":"often we take 0, whatever number is convenient of the same thing."},{"Start":"00:41.060 ","End":"00:45.184","Text":"X over square root of x squared plus 4,"},{"Start":"00:45.184 ","End":"00:51.480","Text":"x over square root of x squared plus 4 and dx of course,"},{"Start":"00:51.480 ","End":"00:55.860","Text":"and both of these have to exist."},{"Start":"00:55.860 ","End":"01:00.830","Text":"Then it will converge and the value will be the sum of these 2."},{"Start":"01:00.830 ","End":"01:04.280","Text":"Let\u0027s just work on one of them and see what happens."},{"Start":"01:04.280 ","End":"01:05.810","Text":"Let\u0027s take the first one,"},{"Start":"01:05.810 ","End":"01:08.755","Text":"the integral from 0 to infinity."},{"Start":"01:08.755 ","End":"01:11.940","Text":"This is equal to the limit."},{"Start":"01:11.940 ","End":"01:15.085","Text":"For infinity, we put a letter like b,"},{"Start":"01:15.085 ","End":"01:18.590","Text":"where b goes to infinity and we replace the infinity"},{"Start":"01:18.590 ","End":"01:22.655","Text":"with b and other than that, everything is unchanged."},{"Start":"01:22.655 ","End":"01:29.330","Text":"X over x squared plus 4 under the square root sign, dx."},{"Start":"01:29.330 ","End":"01:32.795","Text":"I\u0027m just working on this part here."},{"Start":"01:32.795 ","End":"01:35.510","Text":"That will be my asterisk, so to speak."},{"Start":"01:35.510 ","End":"01:37.570","Text":"It\u0027s called an asterisk."},{"Start":"01:37.570 ","End":"01:41.490","Text":"I\u0027m just working on asterisk at the moment."},{"Start":"01:42.470 ","End":"01:47.465","Text":"Now, I need the indefinite integral before I can do the definite integral."},{"Start":"01:47.465 ","End":"01:49.670","Text":"Let\u0027s do that at the side."},{"Start":"01:49.670 ","End":"01:52.220","Text":"That will also be good for the next one."},{"Start":"01:52.220 ","End":"01:56.850","Text":"We might as well find the integral indefinite"},{"Start":"01:56.850 ","End":"02:04.765","Text":"of x over square root of x squared plus 4."},{"Start":"02:04.765 ","End":"02:08.665","Text":"Let\u0027s see. I think a substitution would be in order."},{"Start":"02:08.665 ","End":"02:14.215","Text":"Let\u0027s substitute the square root of x squared plus 4 to be"},{"Start":"02:14.215 ","End":"02:20.910","Text":"t. Take t equals the square root of x squared plus 4,"},{"Start":"02:20.910 ","End":"02:25.280","Text":"and then, dt is equal, let\u0027s see,"},{"Start":"02:25.280 ","End":"02:30.400","Text":"1 over twice the square root of x squared plus 4."},{"Start":"02:30.400 ","End":"02:31.960","Text":"That\u0027s from the square root."},{"Start":"02:31.960 ","End":"02:34.165","Text":"Now we need the inner derivative,"},{"Start":"02:34.165 ","End":"02:39.375","Text":"which is 2x and all this dx."},{"Start":"02:39.375 ","End":"02:48.035","Text":"Luckily, turns out that the 2 cancels with the 2 and I noticed that this is exactly this."},{"Start":"02:48.035 ","End":"02:51.110","Text":"This thing comes out quite simply."},{"Start":"02:51.110 ","End":"02:56.105","Text":"The integral comes out as the integral of just dt."},{"Start":"02:56.105 ","End":"02:58.130","Text":"That\u0027s all that\u0027s left."},{"Start":"02:58.130 ","End":"03:02.090","Text":"This whole thing is just dt and the integral of dt,"},{"Start":"03:02.090 ","End":"03:05.420","Text":"well, dt is just 1 dt."},{"Start":"03:05.420 ","End":"03:10.195","Text":"This is equal to just t plus a constant."},{"Start":"03:10.195 ","End":"03:13.940","Text":"T was the square root of x squared plus 4."},{"Start":"03:13.940 ","End":"03:20.680","Text":"Our answer is the square root of x squared plus 4, plus,"},{"Start":"03:20.680 ","End":"03:22.400","Text":"here I have to put a constant,"},{"Start":"03:22.400 ","End":"03:26.570","Text":"but I can then throw it out here for when I\u0027m doing the definite integrals,"},{"Start":"03:26.570 ","End":"03:28.325","Text":"we don\u0027t need the constant."},{"Start":"03:28.325 ","End":"03:35.770","Text":"We get the limit as b goes to infinity of the square root of x"},{"Start":"03:35.770 ","End":"03:43.305","Text":"squared plus 4 between the limits 0 and b and that equals?"},{"Start":"03:43.305 ","End":"03:47.574","Text":"Let us see. We put in b,"},{"Start":"03:47.574 ","End":"03:52.985","Text":"we get the square root of b squared plus 4."},{"Start":"03:52.985 ","End":"03:58.870","Text":"We put in 0, we get the square root of 4,"},{"Start":"03:58.870 ","End":"04:02.880","Text":"which is just 2, so that\u0027s minus 2."},{"Start":"04:02.880 ","End":"04:11.165","Text":"Then we want the limit as b goes to infinity of this and put it in brackets."},{"Start":"04:11.165 ","End":"04:14.315","Text":"Now, how do we do this kind of a limit?"},{"Start":"04:14.315 ","End":"04:17.915","Text":"I think it\u0027s pretty straightforward to see that this is just infinity,"},{"Start":"04:17.915 ","End":"04:20.300","Text":"because when b goes to infinity,"},{"Start":"04:20.300 ","End":"04:24.350","Text":"b squared goes to infinity and so does b squared plus 4."},{"Start":"04:24.350 ","End":"04:26.630","Text":"The square root of infinity is infinity,"},{"Start":"04:26.630 ","End":"04:29.180","Text":"infinity minus 2 is infinity."},{"Start":"04:29.180 ","End":"04:33.890","Text":"Ultimately, what we get for this is infinity."},{"Start":"04:33.890 ","End":"04:36.790","Text":"Now once we\u0027ve got this asterisk as infinity,"},{"Start":"04:36.790 ","End":"04:39.860","Text":"there\u0027s no need to continue and do the other one."},{"Start":"04:39.860 ","End":"04:44.690","Text":"This just means that this thing diverges."},{"Start":"04:44.690 ","End":"04:46.070","Text":"We don\u0027t need the other part."},{"Start":"04:46.070 ","End":"04:50.315","Text":"Let me just write the answer that this thing diverges."},{"Start":"04:50.315 ","End":"04:52.400","Text":"There\u0027s no limit to compute this,"},{"Start":"04:52.400 ","End":"04:56.700","Text":"there\u0027s no improper integral. We\u0027re done."}],"ID":4590},{"Watched":false,"Name":"Exercise 21","Duration":"12m 5s","ChapterTopicVideoID":4582,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Here we have an improper integral because of"},{"Start":"00:02.280 ","End":"00:06.150","Text":"the infinity and we have to decide if it converges or diverges,"},{"Start":"00:06.150 ","End":"00:08.970","Text":"and if it converges, we evaluate it."},{"Start":"00:08.970 ","End":"00:13.680","Text":"Let\u0027s see how do we handle situation where there\u0027s infinity twice."},{"Start":"00:13.680 ","End":"00:15.000","Text":"We just choose a number,"},{"Start":"00:15.000 ","End":"00:18.225","Text":"I\u0027m going to choose 0 and split it up into 2."},{"Start":"00:18.225 ","End":"00:23.610","Text":"Let\u0027s take it from minus infinity to 0 of the same thing,"},{"Start":"00:23.610 ","End":"00:27.690","Text":"plus the integral from 0 to infinity of the same thing."},{"Start":"00:27.690 ","End":"00:29.290","Text":"Let\u0027s just write it out again,"},{"Start":"00:29.290 ","End":"00:34.334","Text":"e to the minus x over 1 plus twice e to the minus x dx."},{"Start":"00:34.334 ","End":"00:42.725","Text":"Again e to the minus x over 1 plus twice e to the minus x dx."},{"Start":"00:42.725 ","End":"00:45.920","Text":"Each of these has to converge in order for it to converge."},{"Start":"00:45.920 ","End":"00:47.855","Text":"If 1 of them fails we\u0027ve failed."},{"Start":"00:47.855 ","End":"00:53.410","Text":"Let\u0027s start with the first 1 and call this 1 asterisk."},{"Start":"00:53.410 ","End":"00:58.045","Text":"Let\u0027s just do the first 1 asterisk at the side."},{"Start":"00:58.045 ","End":"01:01.565","Text":"How do I handle this minus infinity?"},{"Start":"01:01.565 ","End":"01:05.765","Text":"The way we do this is by letting this be a letter,"},{"Start":"01:05.765 ","End":"01:11.630","Text":"say a, and letting a 10 to minus infinity and take the limit."},{"Start":"01:11.630 ","End":"01:18.080","Text":"What I mean is this asterisk will be the limit of the integral,"},{"Start":"01:18.080 ","End":"01:21.725","Text":"instead of from minus infinity to 0, from a to 0,"},{"Start":"01:21.725 ","End":"01:26.195","Text":"but a goes to minus infinity and have the same thing,"},{"Start":"01:26.195 ","End":"01:33.485","Text":"e to the minus x over 1 plus twice e to the minus x dx."},{"Start":"01:33.485 ","End":"01:35.750","Text":"Now, we\u0027re going to need the indefinite"},{"Start":"01:35.750 ","End":"01:40.115","Text":"integral of this 1 in order to do the definite integral."},{"Start":"01:40.115 ","End":"01:43.160","Text":"Let\u0027s see if we can compute that at the side."},{"Start":"01:43.160 ","End":"01:50.720","Text":"What I want to compute is the integral of e to the minus x over 1"},{"Start":"01:50.720 ","End":"01:58.265","Text":"plus twice e to the minus x. I can tell this 1 would be done best by substitution."},{"Start":"01:58.265 ","End":"02:02.060","Text":"I\u0027m going to let t equals e to"},{"Start":"02:02.060 ","End":"02:07.940","Text":"the minus x. I think I copied the exercise wrong somewhere."},{"Start":"02:07.940 ","End":"02:10.865","Text":"I guess I\u0027m missing some 2s here. Hang on."},{"Start":"02:10.865 ","End":"02:14.090","Text":"Okay, I\u0027ve fixed it, it\u0027s e to the minus 2x,"},{"Start":"02:14.090 ","End":"02:16.610","Text":"and also here and here and here."},{"Start":"02:16.610 ","End":"02:20.555","Text":"Same substitution, t equals e to the minus x,"},{"Start":"02:20.555 ","End":"02:27.770","Text":"and then dt will equal e to the minus x times minus 1,"},{"Start":"02:27.770 ","End":"02:32.850","Text":"which I\u0027ll write in front, times dx."},{"Start":"02:33.040 ","End":"02:35.960","Text":"Let\u0027s see what we have here."},{"Start":"02:35.960 ","End":"02:40.115","Text":"After the substitution, we get the integral."},{"Start":"02:40.115 ","End":"02:43.640","Text":"Now, we don\u0027t have minus e to the minus x."},{"Start":"02:43.640 ","End":"02:45.140","Text":"We have e to the minus x."},{"Start":"02:45.140 ","End":"02:47.270","Text":"But if I put a minus here,"},{"Start":"02:47.270 ","End":"02:48.545","Text":"then it will be okay."},{"Start":"02:48.545 ","End":"02:52.190","Text":"But I have to compensate by also putting a minus here."},{"Start":"02:52.190 ","End":"02:56.390","Text":"Now we can see that we get minus the integral."},{"Start":"02:56.390 ","End":"02:59.990","Text":"Now this whole minus e to the minus x dx is dt."},{"Start":"02:59.990 ","End":"03:04.759","Text":"On the denominator I have 1 plus 2."},{"Start":"03:04.759 ","End":"03:06.680","Text":"But the laws of exponents,"},{"Start":"03:06.680 ","End":"03:10.945","Text":"this is e to the minus x all squared, hence t squared."},{"Start":"03:10.945 ","End":"03:13.805","Text":"This looks like the thing we can do."},{"Start":"03:13.805 ","End":"03:15.860","Text":"I\u0027ll tell you how I\u0027m going to do this."},{"Start":"03:15.860 ","End":"03:23.690","Text":"I\u0027m going to use another formula that the integral of, well,"},{"Start":"03:23.690 ","End":"03:27.785","Text":"it\u0027s written in terms of x and I\u0027ll write it in terms of t. The integral"},{"Start":"03:27.785 ","End":"03:32.660","Text":"of 1 over a squared plus t squared"},{"Start":"03:32.660 ","End":"03:37.170","Text":"dt is equal to 1 over"},{"Start":"03:37.170 ","End":"03:45.020","Text":"a arc tangent of t over a plus constant."},{"Start":"03:45.020 ","End":"03:48.515","Text":"This looks very much like this."},{"Start":"03:48.515 ","End":"03:53.955","Text":"I\u0027m just going to see if I can tweak it a little bit."},{"Start":"03:53.955 ","End":"03:56.655","Text":"Let\u0027s get back there."},{"Start":"03:56.655 ","End":"04:00.345","Text":"If I take the 2 outside,"},{"Start":"04:00.345 ","End":"04:05.580","Text":"I\u0027ll get minus a 1/2 and I\u0027ve taken 2 out."},{"Start":"04:05.580 ","End":"04:08.340","Text":"Here I left with 1.5."},{"Start":"04:08.340 ","End":"04:11.265","Text":"I have dt over."},{"Start":"04:11.265 ","End":"04:13.240","Text":"Instead of writing a 1/2,"},{"Start":"04:13.240 ","End":"04:19.740","Text":"I can write the square root of a 1/2 squared plus t squared."},{"Start":"04:19.740 ","End":"04:22.760","Text":"You can check that this is okay if we multiply out,"},{"Start":"04:22.760 ","End":"04:24.590","Text":"we get 1/2 plus t squared,"},{"Start":"04:24.590 ","End":"04:29.040","Text":"but together with the 2, it gives us 1 plus 2 t squared."},{"Start":"04:29.420 ","End":"04:32.445","Text":"Now I\u0027m going to use the formula,"},{"Start":"04:32.445 ","End":"04:36.815","Text":"and the formula says that this integral is 1 over a."},{"Start":"04:36.815 ","End":"04:40.325","Text":"Let\u0027s keep the 1/2 minus here,"},{"Start":"04:40.325 ","End":"04:44.165","Text":"and what we get is 1 over a,"},{"Start":"04:44.165 ","End":"04:48.110","Text":"but a in this case, is what?"},{"Start":"04:48.110 ","End":"04:49.585","Text":"Square root of 1/2."},{"Start":"04:49.585 ","End":"04:53.230","Text":"Square root of 1/2 can write as 1 over root 2."},{"Start":"04:53.230 ","End":"04:56.685","Text":"It\u0027s minus 1/2. What was here?"},{"Start":"04:56.685 ","End":"04:59.555","Text":"1 over a would be square root of 2,"},{"Start":"04:59.555 ","End":"05:02.555","Text":"because 1 over 1 over this thing itself."},{"Start":"05:02.555 ","End":"05:06.890","Text":"Arc tangent t over a is t over 1"},{"Start":"05:06.890 ","End":"05:11.175","Text":"over square root of 2 is square root of 2t plus constant."},{"Start":"05:11.175 ","End":"05:12.675","Text":"We don\u0027t really need it."},{"Start":"05:12.675 ","End":"05:14.460","Text":"Let\u0027s see if I just simplify it,"},{"Start":"05:14.460 ","End":"05:20.110","Text":"I get square root of 2 over 2 negative."},{"Start":"05:20.690 ","End":"05:26.930","Text":"Arc tangent, I should have put brackets here maybe, square root of 2t."},{"Start":"05:26.930 ","End":"05:29.755","Text":"But we want not t but x,"},{"Start":"05:29.755 ","End":"05:33.500","Text":"so it\u0027s e to the minus x."},{"Start":"05:33.500 ","End":"05:37.570","Text":"That\u0027s the indefinite integral."},{"Start":"05:37.570 ","End":"05:41.030","Text":"Now I\u0027m going to get back here."},{"Start":"05:41.150 ","End":"05:44.200","Text":"Using this indefinite integral,"},{"Start":"05:44.200 ","End":"05:53.265","Text":"I can rewrite this as the limit as a goes to infinity."},{"Start":"05:53.265 ","End":"05:55.360","Text":"Let\u0027s write this thing here."},{"Start":"05:55.360 ","End":"05:58.585","Text":"The minus square root of 2 over 2 can come out front."},{"Start":"05:58.585 ","End":"06:01.810","Text":"Minus square root of 2 over 2 comes out front."},{"Start":"06:01.810 ","End":"06:09.905","Text":"I have arc tangent of square root of 2e to the minus x."},{"Start":"06:09.905 ","End":"06:15.680","Text":"All this taken between a and 0,"},{"Start":"06:15.680 ","End":"06:18.635","Text":"and this is a minus infinity."},{"Start":"06:18.635 ","End":"06:23.360","Text":"What we get is minus the square root of 2 over 2,"},{"Start":"06:23.360 ","End":"06:29.090","Text":"limit, a goes to minus infinity."},{"Start":"06:29.090 ","End":"06:36.710","Text":"If we plug in 0, we get e to the minus 0 as e to the 0 is 1."},{"Start":"06:36.710 ","End":"06:41.785","Text":"We just get arc tangent of square root of"},{"Start":"06:41.785 ","End":"06:52.825","Text":"2 less arc tangent square root of 2e to the minus a."},{"Start":"06:52.825 ","End":"06:55.990","Text":"Now if I put the minus infinity here,"},{"Start":"06:55.990 ","End":"07:04.255","Text":"I get minus the square root of 2 over 2 arc tangent of square root of 2"},{"Start":"07:04.255 ","End":"07:13.780","Text":"minus arc tangent of the square root of 2e to the power of minus minus infinity."},{"Start":"07:13.780 ","End":"07:17.770","Text":"I\u0027m just emphasizing that by writing plus infinity,"},{"Start":"07:17.770 ","End":"07:23.570","Text":"because a is minus infinity and we have minus of minus infinity."},{"Start":"07:24.270 ","End":"07:27.450","Text":"What do we get here?"},{"Start":"07:27.450 ","End":"07:31.970","Text":"We get this square root of 2 times e to the infinity altogether is"},{"Start":"07:31.970 ","End":"07:38.200","Text":"just infinity and the arc-tangent of infinity is Pi over 2."},{"Start":"07:38.200 ","End":"07:44.320","Text":"This bit, I know is Pi over 2 and arc tangent of square root of 2,"},{"Start":"07:44.320 ","End":"07:52.925","Text":"it\u0027s arc tangent of root 2 minus Pi minus square root of 2 over 2."},{"Start":"07:52.925 ","End":"07:58.925","Text":"The first part, the asterisk, certainly did converge."},{"Start":"07:58.925 ","End":"08:01.340","Text":"Let\u0027s see now the second part."},{"Start":"08:01.340 ","End":"08:05.690","Text":"The second part I\u0027ll have to mark with a double asterisk."},{"Start":"08:05.690 ","End":"08:07.730","Text":"This time we get the limit,"},{"Start":"08:07.730 ","End":"08:11.675","Text":"and I\u0027ll call this b, so we\u0027ll have b goes to plus infinity,"},{"Start":"08:11.675 ","End":"08:14.270","Text":"the integral from 0 to b,"},{"Start":"08:14.270 ","End":"08:21.290","Text":"e to the minus x over 1 plus e to the minus 2x dx."},{"Start":"08:21.290 ","End":"08:24.304","Text":"We already have the indefinite integral."},{"Start":"08:24.304 ","End":"08:30.500","Text":"This is equal to square root of 2 minus the square root of 2 over 2."},{"Start":"08:30.500 ","End":"08:33.200","Text":"Take out the limit as before."},{"Start":"08:33.200 ","End":"08:40.435","Text":"Then what we have is the arc tangent of square root of"},{"Start":"08:40.435 ","End":"08:47.810","Text":"2e to the minus x as b goes to infinity."},{"Start":"08:47.810 ","End":"08:52.150","Text":"This is evaluated between 0 and b,"},{"Start":"08:52.150 ","End":"08:54.790","Text":"which means we have to substitute each and subtract."},{"Start":"08:54.790 ","End":"09:00.280","Text":"What we get is minus square root of 2 over 2 times."},{"Start":"09:00.280 ","End":"09:03.790","Text":"Now if I substitute b here,"},{"Start":"09:03.790 ","End":"09:11.765","Text":"I get arc tangent of square root of 2e to the minus b."},{"Start":"09:11.765 ","End":"09:14.335","Text":"If I substitute 0,"},{"Start":"09:14.335 ","End":"09:16.225","Text":"well, we did that already,"},{"Start":"09:16.225 ","End":"09:21.790","Text":"we get arc-tangent of square root of 2."},{"Start":"09:22.340 ","End":"09:25.440","Text":"I forgot the limit but you know what?"},{"Start":"09:25.440 ","End":"09:26.955","Text":"I\u0027ll just do it this way."},{"Start":"09:26.955 ","End":"09:36.210","Text":"I\u0027ll just say here we want the limit as b goes to infinity, just forgot it."},{"Start":"09:36.250 ","End":"09:38.660","Text":"If b goes to infinity,"},{"Start":"09:38.660 ","End":"09:43.415","Text":"then e to the minus b is e to the minus infinity."},{"Start":"09:43.415 ","End":"09:48.559","Text":"It is known that e to the minus infinity is 0,"},{"Start":"09:48.559 ","End":"09:51.050","Text":"so this goes to 0."},{"Start":"09:51.050 ","End":"09:54.320","Text":"What we\u0027re left with is arc tangent of 0."},{"Start":"09:54.320 ","End":"09:56.105","Text":"I\u0027ll write it all properly,"},{"Start":"09:56.105 ","End":"09:59.150","Text":"square root of 2 over 2, arc tangent,"},{"Start":"09:59.150 ","End":"10:09.220","Text":"I\u0027ll write it as arc tangent 0 minus arc tangent square root of 2."},{"Start":"10:09.220 ","End":"10:11.415","Text":"But arc tangent of 0,"},{"Start":"10:11.415 ","End":"10:14.080","Text":"this is equal to 0."},{"Start":"10:18.530 ","End":"10:25.680","Text":"Minus square root of 2 over 2 times minus arc tangent square root of 2."},{"Start":"10:25.680 ","End":"10:31.420","Text":"Allow me to take this minus here together"},{"Start":"10:31.420 ","End":"10:38.225","Text":"with this minus here and combine them together to make a plus."},{"Start":"10:38.225 ","End":"10:43.170","Text":"I\u0027ll just stress that it\u0027s plus because of the 2 minuses."},{"Start":"10:43.170 ","End":"10:47.765","Text":"What I get is basically arc tangent of square root of 2."},{"Start":"10:47.765 ","End":"10:50.670","Text":"This is double asterisk,"},{"Start":"10:50.670 ","End":"10:52.880","Text":"I\u0027ll put it in brackets."},{"Start":"10:52.880 ","End":"11:04.200","Text":"My final answer, which is asterisk plus double asterisk is equal to,"},{"Start":"11:04.200 ","End":"11:11.925","Text":"what we need is this plus arc tangent square root of 2."},{"Start":"11:11.925 ","End":"11:17.795","Text":"Look, this thing is the same as this with minus so these 2 cancel."},{"Start":"11:17.795 ","End":"11:19.985","Text":"I\u0027m saying that when I add them,"},{"Start":"11:19.985 ","End":"11:29.225","Text":"it\u0027s as if this cancels with this and what we\u0027re left with is this times this,"},{"Start":"11:29.225 ","End":"11:34.100","Text":"which will give me a positive number because again it\u0027s a minus with a minus."},{"Start":"11:34.100 ","End":"11:42.260","Text":"I just get minus the square root of 2 over 2 minus Pi over 2."},{"Start":"11:42.260 ","End":"11:45.364","Text":"But the minus with a minus gives me a plus."},{"Start":"11:45.364 ","End":"11:57.560","Text":"What I get is the square root of 2 times Pi over 2 times 2, which is 4."},{"Start":"11:57.560 ","End":"12:00.635","Text":"This is the answer I get."},{"Start":"12:00.635 ","End":"12:06.330","Text":"I hope I didn\u0027t make a mistake and we are done."}],"ID":4591},{"Watched":false,"Name":"Exercise 22","Duration":"9m 8s","ChapterTopicVideoID":4583,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.625","Text":"We have here an improper integral though it\u0027s not immediately clear why and as such,"},{"Start":"00:05.625 ","End":"00:08.220","Text":"it might converge or diverge."},{"Start":"00:08.220 ","End":"00:13.230","Text":"Our task is to decide which and if it converges to compute the value."},{"Start":"00:13.230 ","End":"00:18.105","Text":"Let\u0027s first of all see why this is an improper integral."},{"Start":"00:18.105 ","End":"00:22.425","Text":"Now this function is supposed to go from 1-4."},{"Start":"00:22.425 ","End":"00:24.720","Text":"But what happens if x equals 2,"},{"Start":"00:24.720 ","End":"00:26.700","Text":"which is in this range."},{"Start":"00:26.700 ","End":"00:32.670","Text":"I\u0027m just right side here that x goes from 1-4,"},{"Start":"00:32.670 ","End":"00:35.670","Text":"that\u0027s the domain for the integration."},{"Start":"00:35.670 ","End":"00:39.410","Text":"When x equals 2, we have a problem because then the denominator 0,"},{"Start":"00:39.410 ","End":"00:43.685","Text":"0 to the 2/3 is 0 and we have 1 over 0,"},{"Start":"00:43.685 ","End":"00:48.905","Text":"which means that we have a point x equals 2, which is problematic."},{"Start":"00:48.905 ","End":"00:52.265","Text":"That the function goes to plus or minus infinity."},{"Start":"00:52.265 ","End":"00:55.324","Text":"That makes it a type 2 improper integral,"},{"Start":"00:55.324 ","End":"00:58.760","Text":"so we have to use the appropriate technique."},{"Start":"00:58.760 ","End":"01:02.780","Text":"What I do is break this integral up into 2 integrals,"},{"Start":"01:02.780 ","End":"01:06.185","Text":"from 1-2 and from 2-4."},{"Start":"01:06.185 ","End":"01:07.940","Text":"Let me just get rid of this stuff."},{"Start":"01:07.940 ","End":"01:15.655","Text":"This equals the integral from 1-2 of the same thing,"},{"Start":"01:15.655 ","End":"01:26.015","Text":"1 over x minus 2 to the power of 2/3 plus the integral from 2-4,"},{"Start":"01:26.015 ","End":"01:33.815","Text":"that\u0027s the rest of it also of 1 over x minus 2 to the power of 2/3 dx."},{"Start":"01:33.815 ","End":"01:36.245","Text":"Because 2 was our problem point."},{"Start":"01:36.245 ","End":"01:41.765","Text":"That\u0027s where the function is undefined and unbounded in plus or minus infinity."},{"Start":"01:41.765 ","End":"01:47.315","Text":"Now, both of these have to converge in order for this to converge,"},{"Start":"01:47.315 ","End":"01:52.980","Text":"let\u0027s call the first 1 asterisk and the second 1,"},{"Start":"01:52.980 ","End":"01:56.145","Text":"I\u0027ll call double asterisk."},{"Start":"01:56.145 ","End":"01:58.460","Text":"At the side I\u0027ll first of all,"},{"Start":"01:58.460 ","End":"02:01.085","Text":"try the single asterisk."},{"Start":"02:01.085 ","End":"02:03.780","Text":"If it diverges, I\u0027m finished,"},{"Start":"02:03.780 ","End":"02:05.570","Text":"this thing diverges, if it converges,"},{"Start":"02:05.570 ","End":"02:08.580","Text":"I\u0027ll have to do the next 1 also."},{"Start":"02:09.190 ","End":"02:14.345","Text":"What I have is the integral from"},{"Start":"02:14.345 ","End":"02:23.715","Text":"1-2 of 1 over x minus 2 to the 2/3 dx."},{"Start":"02:23.715 ","End":"02:26.385","Text":"Now, we have a problem at 2."},{"Start":"02:26.385 ","End":"02:32.385","Text":"What we do is we replace the 2 by something that tends to 2,"},{"Start":"02:32.385 ","End":"02:34.590","Text":"but from below, from the left."},{"Start":"02:34.590 ","End":"02:37.405","Text":"What I do is I say we have the limit,"},{"Start":"02:37.405 ","End":"02:42.120","Text":"let\u0027s say b. I\u0027ll use the letter b goes to 2,"},{"Start":"02:42.120 ","End":"02:48.205","Text":"but from below and then I\u0027ll take the integral from 1-b."},{"Start":"02:48.205 ","End":"02:54.185","Text":"That way I\u0027m avoiding the 2 but am getting close to it from below of the same thing."},{"Start":"02:54.185 ","End":"03:03.630","Text":"But I can rewrite this as x minus 2 to the minus 2/3 dx."},{"Start":"03:03.630 ","End":"03:06.065","Text":"This is a straightforward,"},{"Start":"03:06.065 ","End":"03:11.790","Text":"it\u0027s almost immediate integral because it\u0027s an exponent and though it\u0027s not x,"},{"Start":"03:11.790 ","End":"03:14.040","Text":"it\u0027s x minus 2, the anti-derivative is 1,"},{"Start":"03:14.040 ","End":"03:15.150","Text":"so it\u0027s the same thing."},{"Start":"03:15.150 ","End":"03:17.760","Text":"It\u0027s as if I had x to the minus 2/3."},{"Start":"03:17.760 ","End":"03:24.710","Text":"What I get now is limit b goes to 2 from below of this thing here,"},{"Start":"03:24.710 ","End":"03:26.540","Text":"I raise the power by 1,"},{"Start":"03:26.540 ","End":"03:34.615","Text":"that brings me to 1/3 and divide by the new exponent, which is 1/3."},{"Start":"03:34.615 ","End":"03:39.200","Text":"I don\u0027t need the plus C because I\u0027m doing a definite integral."},{"Start":"03:39.200 ","End":"03:44.345","Text":"I\u0027ve got the limit of this taken between 1-b,"},{"Start":"03:44.345 ","End":"03:45.920","Text":"which means I\u0027m going to plug in b,"},{"Start":"03:45.920 ","End":"03:47.915","Text":"plug in 1, subtract the 2."},{"Start":"03:47.915 ","End":"03:51.450","Text":"Now, this is equal to dividing by 1/3 is"},{"Start":"03:51.450 ","End":"03:55.095","Text":"like multiplying by 3 and I can take the 3 outside the limit."},{"Start":"03:55.095 ","End":"04:01.335","Text":"I\u0027ve got 3 times the limit as b goes to 2 from below."},{"Start":"04:01.335 ","End":"04:04.775","Text":"Now let\u0027s see, I have a subtraction here."},{"Start":"04:04.775 ","End":"04:12.675","Text":"I have b minus 2 to the power of 1/3 less,"},{"Start":"04:12.675 ","End":"04:15.065","Text":"see if I put in 1,"},{"Start":"04:15.065 ","End":"04:19.070","Text":"1 minus 2 is minus 1,"},{"Start":"04:19.070 ","End":"04:24.445","Text":"just minus 1 to the power of 1/3."},{"Start":"04:24.445 ","End":"04:33.435","Text":"This is equal to 3 times the limit as b goes to 2 from below,"},{"Start":"04:33.435 ","End":"04:42.020","Text":"I can write the 1/3 as cube root of cube root of b minus 2,"},{"Start":"04:42.020 ","End":"04:46.850","Text":"1 minus 1 to the third is cube root of minus 1 is minus 1,"},{"Start":"04:46.850 ","End":"04:48.930","Text":"this makes it plus 1."},{"Start":"04:48.930 ","End":"04:51.620","Text":"Really all I have to do in this case,"},{"Start":"04:51.620 ","End":"04:55.190","Text":"I can actually substitute b equals 2 and I will get,"},{"Start":"04:55.190 ","End":"05:00.424","Text":"this thing will come 0 because the cube root of 2 minus 2 is 0."},{"Start":"05:00.424 ","End":"05:04.530","Text":"I end up just getting 3 times 1,"},{"Start":"05:04.530 ","End":"05:07.680","Text":"this denote that this whole thing, when b is 2,"},{"Start":"05:07.680 ","End":"05:11.650","Text":"this goes to 0 when b goes to 2,"},{"Start":"05:11.650 ","End":"05:16.815","Text":"so I get 0 plus 1 times 3 is just 3."},{"Start":"05:16.815 ","End":"05:21.545","Text":"That\u0027s the answer for the single asterisk."},{"Start":"05:21.545 ","End":"05:25.620","Text":"Now we have to do the other 1."},{"Start":"05:25.670 ","End":"05:29.570","Text":"The next 1 will be the double asterisk 1,"},{"Start":"05:29.570 ","End":"05:30.920","Text":"which is very similar to this,"},{"Start":"05:30.920 ","End":"05:34.910","Text":"except it\u0027s from 2-4 instead of 1-2."},{"Start":"05:34.910 ","End":"05:37.970","Text":"Let\u0027s see, just copy."},{"Start":"05:37.970 ","End":"05:41.975","Text":"I can copy from here, double asterisk."},{"Start":"05:41.975 ","End":"05:45.755","Text":"Just have to do it like this except from 2-4,"},{"Start":"05:45.755 ","End":"05:48.845","Text":"the integral from 2-4."},{"Start":"05:48.845 ","End":"05:52.880","Text":"Remember 2 is where we have the problem of 1"},{"Start":"05:52.880 ","End":"06:00.360","Text":"over x minus 2 to the power of 2/3 dx."},{"Start":"06:00.650 ","End":"06:04.040","Text":"Once again, we replace it with the limit."},{"Start":"06:04.040 ","End":"06:06.765","Text":"This time the problem is still at 2,"},{"Start":"06:06.765 ","End":"06:09.090","Text":"but that\u0027s now going to be the lower limit,"},{"Start":"06:09.090 ","End":"06:11.135","Text":"so we approach 2 from above."},{"Start":"06:11.135 ","End":"06:13.205","Text":"Basically what I\u0027m saying is,"},{"Start":"06:13.205 ","End":"06:14.780","Text":"instead of taking it from 2,"},{"Start":"06:14.780 ","End":"06:17.390","Text":"I take it from say, a,"},{"Start":"06:17.390 ","End":"06:23.285","Text":"and I take the limit as a goes to 2 from the right this time,"},{"Start":"06:23.285 ","End":"06:27.995","Text":"started from the left because there we were going up to 2 here going down to 2."},{"Start":"06:27.995 ","End":"06:34.035","Text":"As before, it\u0027s the same thing, here it\u0027s 4."},{"Start":"06:34.035 ","End":"06:42.505","Text":"I can use the exponent x minus 2 to the minus 2/3 dx."},{"Start":"06:42.505 ","End":"06:47.820","Text":"The same integral, it\u0027s just limit,"},{"Start":"06:47.820 ","End":"06:50.515","Text":"a goes to 2 from the right."},{"Start":"06:50.515 ","End":"06:55.550","Text":"We already computed the indefinite integral and we said that it\u0027s this over this,"},{"Start":"06:55.550 ","End":"07:01.340","Text":"which we decided was 3 times x minus 2 to the 1/3."},{"Start":"07:01.340 ","End":"07:06.260","Text":"This, we\u0027ve got to take between instead 1 and b this time,"},{"Start":"07:06.260 ","End":"07:10.440","Text":"between a and 4."},{"Start":"07:10.440 ","End":"07:14.430","Text":"Let\u0027s just substitute each of them."},{"Start":"07:14.430 ","End":"07:16.590","Text":"The 3 I can bring out front,"},{"Start":"07:16.590 ","End":"07:22.085","Text":"so I have the limit as a goes to 2 plus."},{"Start":"07:22.085 ","End":"07:24.350","Text":"If I substitute a,"},{"Start":"07:24.350 ","End":"07:30.300","Text":"I get 3 a minus 2."},{"Start":"07:31.330 ","End":"07:33.740","Text":"The cube root, first of all,"},{"Start":"07:33.740 ","End":"07:39.440","Text":"4 minus 2 now see the 3 went to the front and then substituting"},{"Start":"07:39.440 ","End":"07:46.310","Text":"4 and the 1/3 is cube root minus the cube root of when I put a in,"},{"Start":"07:46.310 ","End":"07:54.825","Text":"so it\u0027s a minus 2 and the limit to this as a goes to 2 plus."},{"Start":"07:54.825 ","End":"07:59.000","Text":"There\u0027s no problem in just substituting a equals 2."},{"Start":"07:59.000 ","End":"08:05.745","Text":"What we get is 3 times say cube root of 2."},{"Start":"08:05.745 ","End":"08:07.840","Text":"If a goes to 2,"},{"Start":"08:07.840 ","End":"08:11.710","Text":"then this comes out to be 0,"},{"Start":"08:11.710 ","End":"08:13.520","Text":"just like here we got 0."},{"Start":"08:13.520 ","End":"08:15.770","Text":"This thing goes to 0,"},{"Start":"08:15.770 ","End":"08:19.400","Text":"so it\u0027s just 3 times the cube root of 2."},{"Start":"08:19.400 ","End":"08:22.175","Text":"Now, what I have to do,"},{"Start":"08:22.175 ","End":"08:25.730","Text":"this finishes double asterisk by the way."},{"Start":"08:25.730 ","End":"08:32.075","Text":"All I need to do now is to combine asterisk plus 2 asterisks,"},{"Start":"08:32.075 ","End":"08:35.885","Text":"because remember at the beginning we have asterisk plus 2 asterisks."},{"Start":"08:35.885 ","End":"08:39.024","Text":"Actually, why don\u0027t I continue over here?"},{"Start":"08:39.024 ","End":"08:41.600","Text":"What we get, finally,"},{"Start":"08:41.600 ","End":"08:43.475","Text":"I\u0027ll do it in a different color."},{"Start":"08:43.475 ","End":"08:49.130","Text":"This equals, we got 3 from here and from"},{"Start":"08:49.130 ","End":"08:55.100","Text":"here we got 3 times the cube root of 2."},{"Start":"08:55.100 ","End":"08:57.185","Text":"Could take the 3 outside the brackets."},{"Start":"08:57.185 ","End":"08:58.310","Text":"Not important."},{"Start":"08:58.310 ","End":"09:00.425","Text":"You can do it or don\u0027t do it."},{"Start":"09:00.425 ","End":"09:03.695","Text":"But this is the final answer,"},{"Start":"09:03.695 ","End":"09:05.945","Text":"and we are done."},{"Start":"09:05.945 ","End":"09:08.790","Text":"It converges to this."}],"ID":4592},{"Watched":false,"Name":"Exercise 23","Duration":"4m 13s","ChapterTopicVideoID":4584,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"Here we have an improper integral although it\u0027s not immediately obvious,"},{"Start":"00:04.200 ","End":"00:06.390","Text":"and I\u0027ll explain why in a moment."},{"Start":"00:06.390 ","End":"00:09.915","Text":"If it\u0027s improper, that means it could converge or diverge."},{"Start":"00:09.915 ","End":"00:13.335","Text":"We have to decide which and if it converges to evaluate it."},{"Start":"00:13.335 ","End":"00:15.080","Text":"I\u0027ve copied out here."},{"Start":"00:15.080 ","End":"00:16.950","Text":"Now, why is this improper?"},{"Start":"00:16.950 ","End":"00:18.675","Text":"Well, take a look."},{"Start":"00:18.675 ","End":"00:21.540","Text":"X seemingly goes from 0-2,"},{"Start":"00:21.540 ","End":"00:24.264","Text":"but x equals 1 is a problem."},{"Start":"00:24.264 ","End":"00:26.960","Text":"Because then we have 0 on the denominator,"},{"Start":"00:26.960 ","End":"00:30.005","Text":"and not only is it not defined at x equals 1,"},{"Start":"00:30.005 ","End":"00:32.000","Text":"but when x is near 1,"},{"Start":"00:32.000 ","End":"00:33.710","Text":"this thing is near infinity."},{"Start":"00:33.710 ","End":"00:36.665","Text":"As x gets closer and closer to 1,"},{"Start":"00:36.665 ","End":"00:39.800","Text":"this thing goes to actually plus infinity,"},{"Start":"00:39.800 ","End":"00:42.895","Text":"but plus or minus, it\u0027s unbounded."},{"Start":"00:42.895 ","End":"00:46.285","Text":"That means that it\u0027s type 2 improper integral,"},{"Start":"00:46.285 ","End":"00:49.190","Text":"and the way we tackle it is by breaking up"},{"Start":"00:49.190 ","End":"00:53.175","Text":"the range based on the problem point, which is 1."},{"Start":"00:53.175 ","End":"01:01.350","Text":"I take the integral from 0-1 of this thing and I add to it the integral from 1-2."},{"Start":"01:01.350 ","End":"01:05.469","Text":"I\u0027ll just copying out,1 over x minus 1 squared"},{"Start":"01:05.469 ","End":"01:12.765","Text":"dx and also 1 over x minus 1 squared dx."},{"Start":"01:12.765 ","End":"01:18.590","Text":"Both of these have to converge in order for this thing to converge."},{"Start":"01:18.590 ","End":"01:20.975","Text":"Let\u0027s give them names."},{"Start":"01:20.975 ","End":"01:24.720","Text":"Let\u0027s call this 1 asterisk,"},{"Start":"01:24.720 ","End":"01:28.500","Text":"and this 1 we\u0027ll call double asterisk."},{"Start":"01:28.500 ","End":"01:31.260","Text":"I\u0027ll begin with asterisk,"},{"Start":"01:31.260 ","End":"01:33.575","Text":"and if that diverges,"},{"Start":"01:33.575 ","End":"01:37.910","Text":"then I don\u0027t have to continue but if it converges,"},{"Start":"01:37.910 ","End":"01:40.070","Text":"I mean it, then we need to continue."},{"Start":"01:40.070 ","End":"01:41.795","Text":"Let me copy that."},{"Start":"01:41.795 ","End":"01:45.860","Text":"Just this bit, it\u0027s the integral from 0-1,"},{"Start":"01:45.860 ","End":"01:52.550","Text":"1 over x minus 1 squared dx."},{"Start":"01:52.550 ","End":"01:55.580","Text":"Now, what we do with an improper integral,"},{"Start":"01:55.580 ","End":"01:58.015","Text":"because 1 is a problem,"},{"Start":"01:58.015 ","End":"02:02.435","Text":"we replace the 1 with a limit of something going to 1."},{"Start":"02:02.435 ","End":"02:04.280","Text":"I get the limit,"},{"Start":"02:04.280 ","End":"02:11.660","Text":"let\u0027s say b goes to 1 of the integral from 0 to b of the same thing."},{"Start":"02:11.660 ","End":"02:13.555","Text":"But it\u0027s not exactly 1,"},{"Start":"02:13.555 ","End":"02:17.780","Text":"it\u0027s 1 from below because I\u0027m talking about the range from 0-1,"},{"Start":"02:17.780 ","End":"02:19.190","Text":"so I can\u0027t go above the 1."},{"Start":"02:19.190 ","End":"02:21.395","Text":"It\u0027s 1 with a little minus here,"},{"Start":"02:21.395 ","End":"02:30.280","Text":"goes to 1 from the left or from below of 1 over x minus 1 squared dx."},{"Start":"02:30.590 ","End":"02:33.470","Text":"Now, let\u0027s do the integration."},{"Start":"02:33.470 ","End":"02:36.425","Text":"First, the limit b goes to 1 from below."},{"Start":"02:36.425 ","End":"02:40.190","Text":"Now, the integral of 1 over x squared is minus 1 over x."},{"Start":"02:40.190 ","End":"02:41.780","Text":"We\u0027ve done it so many times."},{"Start":"02:41.780 ","End":"02:43.925","Text":"I\u0027m just going to quote the answer,"},{"Start":"02:43.925 ","End":"02:47.360","Text":"but in our case it\u0027s not x squared is x minus 1 squared,"},{"Start":"02:47.360 ","End":"02:50.705","Text":"but makes no difference because the inner derivative is 1."},{"Start":"02:50.705 ","End":"02:58.350","Text":"This is minus 1 over x minus 1 and we\u0027ll take this from 0-b."},{"Start":"02:58.420 ","End":"03:02.090","Text":"What this means is that I substitute b,"},{"Start":"03:02.090 ","End":"03:04.010","Text":"substitute 0 and subtract."},{"Start":"03:04.010 ","End":"03:07.370","Text":"We\u0027re still under the limit of, let\u0027s see,"},{"Start":"03:07.370 ","End":"03:09.385","Text":"if I put b in here,"},{"Start":"03:09.385 ","End":"03:11.530","Text":"it actually is a little trick I use often."},{"Start":"03:11.530 ","End":"03:12.860","Text":"If it\u0027s a minus,"},{"Start":"03:12.860 ","End":"03:15.110","Text":"then instead of subtracting this from this,"},{"Start":"03:15.110 ","End":"03:16.310","Text":"I subtract this from this,"},{"Start":"03:16.310 ","End":"03:17.975","Text":"and then I can treat it as a plus."},{"Start":"03:17.975 ","End":"03:22.800","Text":"Basically I get 1 over 0 minus 1,"},{"Start":"03:22.800 ","End":"03:27.585","Text":"less 1 over b minus 1."},{"Start":"03:27.585 ","End":"03:31.845","Text":"As b goes to 1 from below,"},{"Start":"03:31.845 ","End":"03:34.909","Text":"and b goes to 1 from below."},{"Start":"03:34.909 ","End":"03:38.240","Text":"That means that this is 1 over 0 minus 1."},{"Start":"03:38.240 ","End":"03:43.860","Text":"This is 1 minus 1 over b slightly under 1,"},{"Start":"03:43.860 ","End":"03:49.790","Text":"and b minus 1 is 0 minus."},{"Start":"03:49.790 ","End":"03:52.310","Text":"That means this over this is minus infinity."},{"Start":"03:52.310 ","End":"03:54.710","Text":"So altogether I get minus minus infinity,"},{"Start":"03:54.710 ","End":"03:56.945","Text":"which is plus infinity."},{"Start":"03:56.945 ","End":"04:00.305","Text":"In any event, it diverges,"},{"Start":"04:00.305 ","End":"04:05.230","Text":"which means that the whole thing diverges."},{"Start":"04:05.230 ","End":"04:08.360","Text":"Because as soon as 1 bit of it diverges,"},{"Start":"04:08.360 ","End":"04:13.800","Text":"then so does the whole thing and that we can end here. We\u0027re done."}],"ID":4593},{"Watched":false,"Name":"Exercise 24","Duration":"3m 33s","ChapterTopicVideoID":4585,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.565","Text":"This integral here is actually an improper integral as we\u0027ll soon see."},{"Start":"00:04.565 ","End":"00:09.390","Text":"That\u0027s why they\u0027re asking us to decide whether it converges or diverges."},{"Start":"00:09.390 ","End":"00:11.850","Text":"If it converges to evaluate it."},{"Start":"00:11.850 ","End":"00:16.020","Text":"The reason is, is that something happens at x equals 2,"},{"Start":"00:16.020 ","End":"00:18.360","Text":"which is in the range from 0-4,"},{"Start":"00:18.360 ","End":"00:20.610","Text":"and x equals 2, it\u0027s not defined,"},{"Start":"00:20.610 ","End":"00:22.139","Text":"but more than that, it\u0027s unbounded."},{"Start":"00:22.139 ","End":"00:24.720","Text":"If x gets nearer and nearer to 2,"},{"Start":"00:24.720 ","End":"00:28.995","Text":"1 over x minus 2 gets close to plus or minus infinity."},{"Start":"00:28.995 ","End":"00:31.920","Text":"This is an improper integral in the way we tackle"},{"Start":"00:31.920 ","End":"00:34.365","Text":"this kind is just to break this integral up,"},{"Start":"00:34.365 ","End":"00:37.725","Text":"to break up the range 0-4 to stop at 2."},{"Start":"00:37.725 ","End":"00:40.740","Text":"What I mean is we\u0027ll get the integral from 0-2 of"},{"Start":"00:40.740 ","End":"00:44.685","Text":"the same thing plus the integral from 2-4."},{"Start":"00:44.685 ","End":"00:48.495","Text":"Each 1 of them is the same x over 2 dx,"},{"Start":"00:48.495 ","End":"00:51.690","Text":"1 over x minus 2 dx."},{"Start":"00:51.690 ","End":"00:55.035","Text":"Let me give them names that\u0027s called the first one"},{"Start":"00:55.035 ","End":"00:58.880","Text":"asterisk and the second 1 double asterisk."},{"Start":"00:58.880 ","End":"01:01.775","Text":"I\u0027m going to compute these separately."},{"Start":"01:01.775 ","End":"01:03.365","Text":"The reason I do that,"},{"Start":"01:03.365 ","End":"01:07.700","Text":"is each of them has to converge in order for this to converge."},{"Start":"01:07.700 ","End":"01:09.950","Text":"But if the first 1 divergence,"},{"Start":"01:09.950 ","End":"01:12.980","Text":"we can stop there already and say divergent."},{"Start":"01:12.980 ","End":"01:14.480","Text":"Let\u0027s go with the first 1,"},{"Start":"01:14.480 ","End":"01:19.085","Text":"the integral from 0-2 of 1 over x minus 2 dx."},{"Start":"01:19.085 ","End":"01:21.470","Text":"The way we tackle the problem point,"},{"Start":"01:21.470 ","End":"01:23.915","Text":"is by replacing it with a different letter,"},{"Start":"01:23.915 ","End":"01:29.475","Text":"say b, and letting b go to 2 as a limit."},{"Start":"01:29.475 ","End":"01:31.840","Text":"But it\u0027s not just 2."},{"Start":"01:31.840 ","End":"01:34.235","Text":"It can actually only go from below."},{"Start":"01:34.235 ","End":"01:37.460","Text":"I\u0027m going from 0 to 2 and going from the left or from below."},{"Start":"01:37.460 ","End":"01:40.550","Text":"It\u0027s actually 2 minus of the same thing,"},{"Start":"01:40.550 ","End":"01:44.005","Text":"1 over x minus 2 dx."},{"Start":"01:44.005 ","End":"01:47.240","Text":"Now, this is equal to the limit."},{"Start":"01:47.240 ","End":"01:49.715","Text":"Again, b goes to 2 from below."},{"Start":"01:49.715 ","End":"01:56.540","Text":"The integral of 1 over x minus 2 is just natural log of x minus 2,"},{"Start":"01:56.540 ","End":"01:58.685","Text":"same as if it was 1 over x."},{"Start":"01:58.685 ","End":"02:05.735","Text":"But this has to be evaluated between 0 and b,"},{"Start":"02:05.735 ","End":"02:08.555","Text":"which means that we have to substitute b,"},{"Start":"02:08.555 ","End":"02:10.760","Text":"substitute 0 and subtract."},{"Start":"02:10.760 ","End":"02:18.350","Text":"We get again limit of natural log of b minus"},{"Start":"02:18.350 ","End":"02:27.095","Text":"2 and absolute value minus natural log of 0 minus 2 is just natural log of 2."},{"Start":"02:27.095 ","End":"02:31.030","Text":"B goes to 2 from below."},{"Start":"02:31.360 ","End":"02:36.065","Text":"What this basically equals is,"},{"Start":"02:36.065 ","End":"02:39.740","Text":"if we basically just let b go to 2 from below,"},{"Start":"02:39.740 ","End":"02:46.340","Text":"we can actually symbolically say that this is natural log of 2 minus,"},{"Start":"02:46.340 ","End":"02:52.560","Text":"minus 2 minus natural log of 2."},{"Start":"02:52.560 ","End":"02:54.650","Text":"Now, 2 minus, minus 2,"},{"Start":"02:54.650 ","End":"03:01.710","Text":"doing all this infinitesimal arithmetic is just 0 minus just something slightly below 2,"},{"Start":"03:01.710 ","End":"03:05.440","Text":"less 2 is something slightly below 0."},{"Start":"03:05.780 ","End":"03:11.630","Text":"This is just short for the limit of natural log of x as x goes to 0."},{"Start":"03:11.630 ","End":"03:15.905","Text":"This thing is known to be minus infinity."},{"Start":"03:15.905 ","End":"03:21.435","Text":"What we get is that this thing diverges."},{"Start":"03:21.435 ","End":"03:25.835","Text":"If the first 1 diverges and the whole thing diverges,"},{"Start":"03:25.835 ","End":"03:27.470","Text":"so we don\u0027t have to go anymore."},{"Start":"03:27.470 ","End":"03:34.620","Text":"We just say this integral is improper integral diverges and we are done."}],"ID":4594},{"Watched":false,"Name":"Exercise 25","Duration":"11m 41s","ChapterTopicVideoID":4586,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Here we have an improper integral."},{"Start":"00:02.550 ","End":"00:05.730","Text":"We can immediately see this because of the infinity,"},{"Start":"00:05.730 ","End":"00:09.510","Text":"we have to decide if it converges or diverges."},{"Start":"00:09.510 ","End":"00:12.750","Text":"If it converges, we have to evaluate it."},{"Start":"00:12.750 ","End":"00:20.190","Text":"Now, I want to point out that it\u0027s not just improper because of the infinity."},{"Start":"00:20.190 ","End":"00:22.680","Text":"But if you look at the other end, the 1,"},{"Start":"00:22.680 ","End":"00:25.240","Text":"we have a problem here too,"},{"Start":"00:25.520 ","End":"00:28.350","Text":"I forgot the dividing line here."},{"Start":"00:28.350 ","End":"00:33.095","Text":"Sorry. The denominator has 2 points, well,"},{"Start":"00:33.095 ","End":"00:36.035","Text":"has several points where it\u0027s undefined,"},{"Start":"00:36.035 ","End":"00:38.390","Text":"0 and plus or minus 1,"},{"Start":"00:38.390 ","End":"00:39.425","Text":"but in our range,"},{"Start":"00:39.425 ","End":"00:41.545","Text":"only 1 is a problem."},{"Start":"00:41.545 ","End":"00:48.135","Text":"This is a problem because 1 squared minus 1 is 0 and then we get 0 in the denominator."},{"Start":"00:48.135 ","End":"00:54.155","Text":"What we have to do in this case is to break it up first of all into 2 pieces."},{"Start":"00:54.155 ","End":"00:56.630","Text":"Choose some point between 1 and infinity."},{"Start":"00:56.630 ","End":"00:59.795","Text":"I\u0027m going to choose 2 and break it up."},{"Start":"00:59.795 ","End":"01:04.385","Text":"We\u0027ll have, say, the integral from 1-2 of whatever this is,"},{"Start":"01:04.385 ","End":"01:08.050","Text":"plus the integral from 2 to infinity."},{"Start":"01:08.050 ","End":"01:09.765","Text":"Just let me write it in,"},{"Start":"01:09.765 ","End":"01:16.065","Text":"1 over x root x squared minus 1 dx."},{"Start":"01:16.065 ","End":"01:22.970","Text":"Same thing here, 1 over x root x squared minus 1 dx."},{"Start":"01:22.970 ","End":"01:28.040","Text":"Now what I\u0027m going to do is label them because they\u0027re going to do each 1 separately."},{"Start":"01:28.040 ","End":"01:35.660","Text":"This 1 will be asterisk and this integral will be double asterisk, put in brackets."},{"Start":"01:35.660 ","End":"01:38.480","Text":"We\u0027re going to do asterisk first."},{"Start":"01:38.480 ","End":"01:43.565","Text":"I want to point out that only if both of these converge, does this converge."},{"Start":"01:43.565 ","End":"01:47.670","Text":"If the first 1 happens to diverge, I can stop there."},{"Start":"01:48.070 ","End":"01:51.095","Text":"Let\u0027s get to the integral."},{"Start":"01:51.095 ","End":"01:54.430","Text":"I want the integral from 1-2,"},{"Start":"01:54.430 ","End":"02:03.535","Text":"just copying 1 over x square root of x squared minus 1 dx."},{"Start":"02:03.535 ","End":"02:08.600","Text":"The first problem is to find the indefinite integral of this, not immediately obvious."},{"Start":"02:08.600 ","End":"02:10.370","Text":"It\u0027s not an immediate integral."},{"Start":"02:10.370 ","End":"02:13.280","Text":"Let\u0027s do that at the side."},{"Start":"02:13.280 ","End":"02:17.100","Text":"The indefinite integral of,"},{"Start":"02:17.100 ","End":"02:24.080","Text":"let\u0027s call it dx over x root x squared minus 1,"},{"Start":"02:24.080 ","End":"02:28.100","Text":"I think the obvious thing to try is a substitution."},{"Start":"02:28.100 ","End":"02:34.430","Text":"Let\u0027s try that in t equals the square root of x squared minus 1."},{"Start":"02:34.430 ","End":"02:42.330","Text":"Then dt will equal 1 over twice the square root because of the square root."},{"Start":"02:42.330 ","End":"02:44.090","Text":"Then the internal derivative,"},{"Start":"02:44.090 ","End":"02:46.910","Text":"which will be 2x."},{"Start":"02:46.910 ","End":"02:50.990","Text":"This 2 with this 2 will cancel."},{"Start":"02:50.990 ","End":"02:55.275","Text":"That\u0027s dt if I need it, of course dx."},{"Start":"02:55.275 ","End":"03:01.190","Text":"In case I need it the other way around dx will be the square root of x squared minus"},{"Start":"03:01.190 ","End":"03:08.300","Text":"1 over x dt by just throwing this fraction to the other side and inverting it."},{"Start":"03:08.300 ","End":"03:10.400","Text":"Let\u0027s see what we have here."},{"Start":"03:10.400 ","End":"03:16.235","Text":"We have the integral of dx is,"},{"Start":"03:16.235 ","End":"03:17.915","Text":"well, let\u0027s start elsewhere."},{"Start":"03:17.915 ","End":"03:19.385","Text":"I want to put this at the end."},{"Start":"03:19.385 ","End":"03:26.375","Text":"I have 1 over x and then I have t. Then I have the dx,"},{"Start":"03:26.375 ","End":"03:29.930","Text":"which is dt times"},{"Start":"03:29.930 ","End":"03:34.760","Text":"the square root of x squared minus 1 over x. I want to just put that in here."},{"Start":"03:34.760 ","End":"03:44.890","Text":"Just let me erase the 1 here and I\u0027ll get to the square root of x squared minus 1."},{"Start":"03:44.890 ","End":"03:49.950","Text":"I got that part and I still need an x times x."},{"Start":"03:49.950 ","End":"03:53.120","Text":"Let\u0027s see. Now what do we get here?"},{"Start":"03:53.120 ","End":"03:55.550","Text":"We\u0027ve got x\u0027s we\u0027ve got to get rid of the x\u0027s."},{"Start":"03:55.550 ","End":"04:03.000","Text":"For 1 thing, this x squared minus 1 under the root sign is t. This is the"},{"Start":"04:03.000 ","End":"04:11.895","Text":"integral of t over x squared times t dt."},{"Start":"04:11.895 ","End":"04:19.280","Text":"Now this t cancels with this t and x squared."},{"Start":"04:19.280 ","End":"04:22.700","Text":"I could probably get that from here."},{"Start":"04:22.700 ","End":"04:27.205","Text":"Yes. What I can say is that from here,"},{"Start":"04:27.205 ","End":"04:30.450","Text":"t squared is x squared minus 1."},{"Start":"04:30.450 ","End":"04:35.465","Text":"X squared is equal to t squared plus 1."},{"Start":"04:35.465 ","End":"04:37.475","Text":"That\u0027s easy algebra, check it."},{"Start":"04:37.475 ","End":"04:43.195","Text":"What I get is the integral"},{"Start":"04:43.195 ","End":"04:51.615","Text":"of 1 over t squared plus 1, dt."},{"Start":"04:51.615 ","End":"04:55.640","Text":"This is an immediate integral far as I remember,"},{"Start":"04:55.640 ","End":"04:57.560","Text":"it\u0027s the arc tangent."},{"Start":"04:57.560 ","End":"05:05.840","Text":"Ultimately we get the arc tangent of t plus c,"},{"Start":"05:05.840 ","End":"05:09.470","Text":"but we don\u0027t really need the c because we\u0027re going to do a definite integral."},{"Start":"05:09.470 ","End":"05:11.600","Text":"Then I have to at the end,"},{"Start":"05:11.600 ","End":"05:15.140","Text":"replace t with whatever it was,"},{"Start":"05:15.140 ","End":"05:17.495","Text":"square root of x squared plus 1."},{"Start":"05:17.495 ","End":"05:20.860","Text":"I\u0027ll finally just get 1 more step here."},{"Start":"05:20.860 ","End":"05:25.880","Text":"This is the arc tangent of"},{"Start":"05:25.880 ","End":"05:32.870","Text":"square root of x squared minus 1 plus the c, which I don\u0027t need."},{"Start":"05:32.870 ","End":"05:35.735","Text":"All I have to do now is remember this."},{"Start":"05:35.735 ","End":"05:41.410","Text":"I pasted it so that when I scroll up again and I\u0027ll see what it is."},{"Start":"05:41.660 ","End":"05:45.485","Text":"Back to the asterisk."},{"Start":"05:45.485 ","End":"05:50.689","Text":"Continuing there, I get the integral becomes"},{"Start":"05:50.689 ","End":"05:58.985","Text":"just arc tangent of square root of x squared minus 1."},{"Start":"05:58.985 ","End":"06:01.010","Text":"I forgot a step."},{"Start":"06:01.010 ","End":"06:03.980","Text":"I forgot to say that because the 1 is not valid,"},{"Start":"06:03.980 ","End":"06:05.210","Text":"it\u0027s not defined at 1,"},{"Start":"06:05.210 ","End":"06:06.695","Text":"1 is a problem point."},{"Start":"06:06.695 ","End":"06:12.510","Text":"That what we do is we take the limit as something goes to 1."},{"Start":"06:12.510 ","End":"06:17.660","Text":"Let\u0027s say a goes to 1 of the integral from a to 2 of"},{"Start":"06:17.660 ","End":"06:26.135","Text":"1 over x root x squared minus 1, dx."},{"Start":"06:26.135 ","End":"06:27.790","Text":"It\u0027s not too late."},{"Start":"06:27.790 ","End":"06:38.510","Text":"Here I just have to put the limit as a goes to 1."},{"Start":"06:38.510 ","End":"06:41.780","Text":"But notice that it\u0027s not just a goes to 1,"},{"Start":"06:41.780 ","End":"06:44.320","Text":"a has to be between 1 and 2."},{"Start":"06:44.320 ","End":"06:46.725","Text":"It goes to 1 from above,"},{"Start":"06:46.725 ","End":"06:49.935","Text":"so it\u0027s 1 plus really."},{"Start":"06:49.935 ","End":"06:55.225","Text":"This taken between the limits of a and 2."},{"Start":"06:55.225 ","End":"06:58.740","Text":"This is equal 2, just plug in 2,"},{"Start":"06:58.740 ","End":"07:00.860","Text":"plug in a, and subtract."},{"Start":"07:00.860 ","End":"07:05.690","Text":"But I still have a limit because a goes to 1 from above."},{"Start":"07:05.690 ","End":"07:11.940","Text":"If I put in 2, I get arc tangent 2 squared minus 1 is 3."},{"Start":"07:11.940 ","End":"07:20.895","Text":"Arc tangent square root of 3 and subtract in the brackets,"},{"Start":"07:20.895 ","End":"07:27.725","Text":"arc tangent of square root of a squared minus 1."},{"Start":"07:27.725 ","End":"07:33.105","Text":"Very good. Now, if a goes to 1 from the right,"},{"Start":"07:33.105 ","End":"07:35.300","Text":"the arc tangent is just a constant,"},{"Start":"07:35.300 ","End":"07:40.055","Text":"so it\u0027s there, whatever arc tangent square root of 3."},{"Start":"07:40.055 ","End":"07:43.910","Text":"Here if a goes to 1 from whatever direction,"},{"Start":"07:43.910 ","End":"07:46.730","Text":"a squared minus 1 goes to 0,"},{"Start":"07:46.730 ","End":"07:51.015","Text":"and so does the square root."},{"Start":"07:51.015 ","End":"07:55.070","Text":"We get arc tangent of 0."},{"Start":"07:55.070 ","End":"07:58.040","Text":"Now I know these things better in degrees."},{"Start":"07:58.040 ","End":"08:00.260","Text":"I\u0027ll just write it above."},{"Start":"08:00.260 ","End":"08:02.990","Text":"I know that the arc tangent is 0 degrees,"},{"Start":"08:02.990 ","End":"08:04.970","Text":"this is 60 degrees,"},{"Start":"08:04.970 ","End":"08:07.070","Text":"and this is 0 degrees,"},{"Start":"08:07.070 ","End":"08:08.240","Text":"but we don\u0027t use degrees,"},{"Start":"08:08.240 ","End":"08:12.350","Text":"we use radians so that we see 60 degrees is 180 over 3,"},{"Start":"08:12.350 ","End":"08:14.665","Text":"that\u0027s pi over 3."},{"Start":"08:14.665 ","End":"08:17.235","Text":"Arc tangent of 0 is 0."},{"Start":"08:17.235 ","End":"08:22.460","Text":"We\u0027ve already computed the first 1 which does converge,"},{"Start":"08:22.460 ","End":"08:24.935","Text":"and that\u0027s the 1 we called asterisk."},{"Start":"08:24.935 ","End":"08:28.990","Text":"Now let\u0027s get on to double asterisk."},{"Start":"08:28.990 ","End":"08:33.900","Text":"This was the integral from 2 to infinity."},{"Start":"08:33.900 ","End":"08:36.800","Text":"Here we go with the next 1,"},{"Start":"08:36.800 ","End":"08:43.370","Text":"the integral from 2 to infinity of the same 1 over"},{"Start":"08:43.370 ","End":"08:51.105","Text":"x square root of x squared minus 1, dx."},{"Start":"08:51.105 ","End":"08:53.625","Text":"This will equal."},{"Start":"08:53.625 ","End":"08:59.435","Text":"This time Infinity is the bad point. It\u0027s not even a point."},{"Start":"08:59.435 ","End":"09:00.950","Text":"What we do here is again,"},{"Start":"09:00.950 ","End":"09:02.735","Text":"use the limit process."},{"Start":"09:02.735 ","End":"09:04.910","Text":"This time I\u0027ll use the letter b."},{"Start":"09:04.910 ","End":"09:07.750","Text":"B goes to infinity."},{"Start":"09:07.750 ","End":"09:13.600","Text":"We get the integral from 2 to b of 1"},{"Start":"09:13.600 ","End":"09:20.455","Text":"over x square root of x squared minus 1, dx."},{"Start":"09:20.455 ","End":"09:24.250","Text":"Well, we\u0027ve already computed the indefinite integral."},{"Start":"09:24.250 ","End":"09:25.870","Text":"Just have to copy it from here,"},{"Start":"09:25.870 ","End":"09:29.650","Text":"but I also need to scroll down a bit. Let\u0027s see."},{"Start":"09:29.650 ","End":"09:37.485","Text":"In our case we have the limit as b goes to infinity,"},{"Start":"09:37.485 ","End":"09:45.850","Text":"now this thing is arc tangent of square root of x squared minus 1."},{"Start":"09:45.850 ","End":"09:49.135","Text":"But this time not from a to 2,"},{"Start":"09:49.135 ","End":"09:52.480","Text":"but in our case it\u0027s from 2 to b."},{"Start":"09:52.480 ","End":"09:55.790","Text":"It proceeds similarly to before."},{"Start":"09:55.790 ","End":"10:00.105","Text":"What we get is the limit,"},{"Start":"10:00.105 ","End":"10:02.475","Text":"b goes to infinity again,"},{"Start":"10:02.475 ","End":"10:09.665","Text":"of arc tangent of the square root of b squared minus 1,"},{"Start":"10:09.665 ","End":"10:16.730","Text":"less arc tangent of 2 squared minus 1,"},{"Start":"10:16.730 ","End":"10:19.320","Text":"which is square root of 3."},{"Start":"10:19.320 ","End":"10:27.065","Text":"This time the limit as b goes to infinity is arc tangent infinity."},{"Start":"10:27.065 ","End":"10:28.265","Text":"Basically, I\u0027ll just write it out."},{"Start":"10:28.265 ","End":"10:33.080","Text":"It\u0027s arc tangent of plus"},{"Start":"10:33.080 ","End":"10:41.260","Text":"infinity minus arc tangent of square root of 3."},{"Start":"10:41.260 ","End":"10:45.025","Text":"The arc tangent of infinity is 90 degrees,"},{"Start":"10:45.025 ","End":"10:47.355","Text":"I still think in degrees."},{"Start":"10:47.355 ","End":"10:50.775","Text":"Let\u0027s make that pi over 2."},{"Start":"10:50.775 ","End":"10:54.564","Text":"This we already computed before is pi over 3,"},{"Start":"10:54.564 ","End":"10:57.474","Text":"need not to have bothered because it cancels anyway."},{"Start":"10:57.474 ","End":"11:02.890","Text":"As you\u0027ll see, we\u0027re going to add what we have now is double asterisk."},{"Start":"11:02.890 ","End":"11:06.820","Text":"Not even going to do the subtraction and say that it\u0027s pi over 6."},{"Start":"11:06.820 ","End":"11:16.275","Text":"Because what I\u0027ll get is if I take the asterisk plus the double asterisk. You know what?"},{"Start":"11:16.275 ","End":"11:18.285","Text":"I can do that up there."},{"Start":"11:18.285 ","End":"11:22.320","Text":"This was equal to pi."},{"Start":"11:22.320 ","End":"11:25.930","Text":"The first 1 was pi over 3,"},{"Start":"11:25.930 ","End":"11:32.930","Text":"and the second 1 was pi over 2 minus pi over 3."},{"Start":"11:32.930 ","End":"11:38.075","Text":"Altogether we get just pi over 2."},{"Start":"11:38.075 ","End":"11:41.970","Text":"That is the answer. Done."}],"ID":4595},{"Watched":false,"Name":"Exercise 26","Duration":"3m 44s","ChapterTopicVideoID":4587,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"Here we have an improper integral,"},{"Start":"00:01.980 ","End":"00:04.035","Text":"and I\u0027ll show you why in a moment."},{"Start":"00:04.035 ","End":"00:10.260","Text":"That\u0027s why they ask us to decide with the following integral converges or diverges."},{"Start":"00:10.260 ","End":"00:12.990","Text":"If it converges, we have to compute it."},{"Start":"00:12.990 ","End":"00:18.720","Text":"Now let\u0027s see why the limits of integration means that x is going from 0 to 1."},{"Start":"00:18.720 ","End":"00:21.720","Text":"But the function is not defined overall this range,"},{"Start":"00:21.720 ","End":"00:25.860","Text":"because look, 0 it\u0027s not defined because I\u0027ve got 0 in the denominator."},{"Start":"00:25.860 ","End":"00:27.780","Text":"Not only is it not defined,"},{"Start":"00:27.780 ","End":"00:32.970","Text":"but it\u0027s also unbounded because sine keeps going from 1 to minus 1,"},{"Start":"00:32.970 ","End":"00:35.670","Text":"but this bit goes to infinity."},{"Start":"00:35.670 ","End":"00:39.230","Text":"All in all it\u0027s unbounded and undefined at x is 0,"},{"Start":"00:39.230 ","End":"00:41.460","Text":"makes it an improper integral of type 2."},{"Start":"00:41.460 ","End":"00:44.930","Text":"The way we deal with that is instead of the 0,"},{"Start":"00:44.930 ","End":"00:47.860","Text":"we take something that tends to 0, let\u0027s say a,"},{"Start":"00:47.860 ","End":"00:51.710","Text":"and we take the limit as a goes to 0,"},{"Start":"00:51.710 ","End":"00:54.875","Text":"but not just 0, but 0 plus,"},{"Start":"00:54.875 ","End":"00:56.360","Text":"because we can only go from the right,"},{"Start":"00:56.360 ","End":"00:57.740","Text":"we\u0027re between 0 and 1."},{"Start":"00:57.740 ","End":"00:58.985","Text":"Everything else is the same,"},{"Start":"00:58.985 ","End":"01:07.335","Text":"1 sine of 1 over x times 1 over x squared dx."},{"Start":"01:07.335 ","End":"01:13.250","Text":"Next thing we need to do is find the indefinite integral of this, the antiderivative."},{"Start":"01:13.250 ","End":"01:16.010","Text":"Let\u0027s do this at the side in another color."},{"Start":"01:16.010 ","End":"01:20.840","Text":"What I want is the integral of"},{"Start":"01:20.840 ","End":"01:28.355","Text":"sine of 1 over x times 1 over x squared dx."},{"Start":"01:28.355 ","End":"01:31.925","Text":"I suggest a substitution."},{"Start":"01:31.925 ","End":"01:38.780","Text":"A substitution, let\u0027s say of t equals 1 over x seems the most natural thing to do."},{"Start":"01:38.780 ","End":"01:43.265","Text":"Then dt is equal to derivative of this."},{"Start":"01:43.265 ","End":"01:45.170","Text":"Well, this is well-known, we\u0027ve done it enough times."},{"Start":"01:45.170 ","End":"01:49.165","Text":"It\u0027s minus 1 over x squared, and that\u0027s dx."},{"Start":"01:49.165 ","End":"01:52.170","Text":"If I had a minus here that would really suit me."},{"Start":"01:52.170 ","End":"01:55.985","Text":"So how about I add a minus here and a minus here?"},{"Start":"01:55.985 ","End":"01:57.890","Text":"That\u0027ll make it okay."},{"Start":"01:57.890 ","End":"02:03.590","Text":"Now what we get is minus sine of t,"},{"Start":"02:03.590 ","End":"02:05.030","Text":"which is 1 over x,"},{"Start":"02:05.030 ","End":"02:07.800","Text":"and all this bit is dt."},{"Start":"02:08.200 ","End":"02:11.270","Text":"The minus sine t, well,"},{"Start":"02:11.270 ","End":"02:12.890","Text":"we know the integral of that,"},{"Start":"02:12.890 ","End":"02:14.770","Text":"that in fact is cosine."},{"Start":"02:14.770 ","End":"02:20.420","Text":"We get cosine t plus a constant,"},{"Start":"02:20.420 ","End":"02:25.460","Text":"and ultimately we just replace t back to 1 over x,"},{"Start":"02:25.460 ","End":"02:33.335","Text":"so we get cosine of 1 over x plus C. We don\u0027t really need the C,"},{"Start":"02:33.335 ","End":"02:37.420","Text":"but I\u0027m going to highlight this just so we\u0027ll use that later,"},{"Start":"02:37.420 ","End":"02:39.400","Text":"the cosine of 1 over x."},{"Start":"02:39.400 ","End":"02:42.130","Text":"Now I\u0027m going to get back here."},{"Start":"02:42.130 ","End":"02:49.570","Text":"We get limit a goes to 0 from above of"},{"Start":"02:49.570 ","End":"02:59.345","Text":"cosine of 1 over x between the limits a and 1."},{"Start":"02:59.345 ","End":"03:05.635","Text":"This equals still the limit a goes to 0 plus,"},{"Start":"03:05.635 ","End":"03:08.545","Text":"plugging 1, plugging a and subtract."},{"Start":"03:08.545 ","End":"03:18.435","Text":"We get cosine of 1 minus cosine of 1 over a, the limit."},{"Start":"03:18.435 ","End":"03:21.515","Text":"Now, cosine of 1 is a constant."},{"Start":"03:21.515 ","End":"03:23.975","Text":"What is cosine 1 over a?"},{"Start":"03:23.975 ","End":"03:28.790","Text":"In fact, this oscillates as we go to 0 all the time between plus and minus 1."},{"Start":"03:28.790 ","End":"03:33.930","Text":"So it actually does not have a limit as a goes to 0."},{"Start":"03:33.930 ","End":"03:35.520","Text":"It doesn\u0027t have a limit."},{"Start":"03:35.520 ","End":"03:45.580","Text":"This one has no limit and therefore this integral diverges. We are done."}],"ID":4596},{"Watched":false,"Name":"Exercise 27","Duration":"10m 17s","ChapterTopicVideoID":4588,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"Here, we have an improper integral and I\u0027ll soon show you why."},{"Start":"00:04.065 ","End":"00:06.960","Text":"As such, it could converge or diverge."},{"Start":"00:06.960 ","End":"00:08.550","Text":"That\u0027s what we have to decide."},{"Start":"00:08.550 ","End":"00:12.390","Text":"If it converges, then we also have to evaluate it."},{"Start":"00:12.390 ","End":"00:14.205","Text":"I\u0027ve written it down here."},{"Start":"00:14.205 ","End":"00:15.840","Text":"Now, what\u0027s the problem?"},{"Start":"00:15.840 ","End":"00:19.995","Text":"The problem is that x goes from 0-1."},{"Start":"00:19.995 ","End":"00:22.785","Text":"But when x is 0,"},{"Start":"00:22.785 ","End":"00:28.875","Text":"the function is undefined because it has a 0 in the denominator."},{"Start":"00:28.875 ","End":"00:34.245","Text":"What we do in this case and this is a type 2 improper integral,"},{"Start":"00:34.245 ","End":"00:38.994","Text":"is we replace the 0 by something that tends to 0."},{"Start":"00:38.994 ","End":"00:44.190","Text":"I\u0027ll call it A, and let A go to 0"},{"Start":"00:44.190 ","End":"00:47.120","Text":"But not just 0 because I\u0027m between 0 and 1,"},{"Start":"00:47.120 ","End":"00:50.350","Text":"I have to go to 0 from above or from the right."},{"Start":"00:50.350 ","End":"00:52.185","Text":"That\u0027s how we write it."},{"Start":"00:52.185 ","End":"00:56.565","Text":"Everything else is the same; 1 over x,"},{"Start":"00:56.565 ","End":"01:03.040","Text":"square root of x squared plus 1, dx."},{"Start":"01:03.230 ","End":"01:05.990","Text":"We\u0027re going to need the indefinite integral"},{"Start":"01:05.990 ","End":"01:07.820","Text":"of this before we can do the definite integral."},{"Start":"01:07.820 ","End":"01:09.275","Text":"Let\u0027s do it at the side."},{"Start":"01:09.275 ","End":"01:15.094","Text":"What I want is the indefinite integral of 1 over x,"},{"Start":"01:15.094 ","End":"01:21.100","Text":"square root of x squared plus 1, dx."},{"Start":"01:21.100 ","End":"01:23.525","Text":"A substitution should do it."},{"Start":"01:23.525 ","End":"01:31.570","Text":"Let\u0027s try t equals the square root of x squared plus 1."},{"Start":"01:31.570 ","End":"01:33.420","Text":"Just in case I need it,"},{"Start":"01:33.420 ","End":"01:35.390","Text":"let\u0027s do the reverse substitution."},{"Start":"01:35.390 ","End":"01:37.670","Text":"If I raise this to the power of 2,"},{"Start":"01:37.670 ","End":"01:40.730","Text":"I get t squared is x squared plus 1,"},{"Start":"01:40.730 ","End":"01:44.410","Text":"t squared minus 1 equals x squared."},{"Start":"01:44.410 ","End":"01:47.790","Text":"We get the square root of t squared minus 1,"},{"Start":"01:47.790 ","End":"01:49.620","Text":"in case I need the reverse."},{"Start":"01:49.620 ","End":"01:53.170","Text":"Now, let\u0027s go for the derivative dt."},{"Start":"01:53.170 ","End":"01:58.430","Text":"It\u0027s 1 over twice the square root of x"},{"Start":"01:58.430 ","End":"02:04.045","Text":"squared plus 1 because that\u0027s the square root and the inner derivative is 2x."},{"Start":"02:04.045 ","End":"02:07.050","Text":"I\u0027m lucky here, it cancels anyway,"},{"Start":"02:07.050 ","End":"02:09.880","Text":"the 2s, so it\u0027s a little bit simpler."},{"Start":"02:14.090 ","End":"02:17.850","Text":"I forgot to write the dx here."},{"Start":"02:17.850 ","End":"02:19.535","Text":"If I want the reverse,"},{"Start":"02:19.535 ","End":"02:25.310","Text":"I can say that dx is equal to square root"},{"Start":"02:25.310 ","End":"02:31.760","Text":"of x squared plus 1 over x."},{"Start":"02:31.760 ","End":"02:34.745","Text":"Just bringing this thing to the other side, dt."},{"Start":"02:34.745 ","End":"02:36.995","Text":"We\u0027ve got all the combinations,"},{"Start":"02:36.995 ","End":"02:39.425","Text":"let\u0027s now do the substitution."},{"Start":"02:39.425 ","End":"02:42.125","Text":"What I get is,"},{"Start":"02:42.125 ","End":"02:44.600","Text":"let\u0027s start with the easiest."},{"Start":"02:44.600 ","End":"02:48.830","Text":"The easiest is the x squared plus 1 here,"},{"Start":"02:48.830 ","End":"02:53.520","Text":"that is t and dx."},{"Start":"02:53.520 ","End":"02:55.845","Text":"I can use that."},{"Start":"02:55.845 ","End":"03:00.930","Text":"I\u0027ll leave the x from there here, that\u0027s this x."},{"Start":"03:00.930 ","End":"03:09.000","Text":"Now, this over this times dt instead of dx or I can put the dt. Let\u0027s take each one."},{"Start":"03:09.000 ","End":"03:11.630","Text":"I\u0027ll put the x over here and I\u0027ll"},{"Start":"03:11.630 ","End":"03:15.220","Text":"put the square root of x squared plus 1 in the numerator."},{"Start":"03:15.220 ","End":"03:17.085","Text":"They can hold the bits;"},{"Start":"03:17.085 ","End":"03:20.970","Text":"this one here, this one here, and this one here."},{"Start":"03:21.620 ","End":"03:29.900","Text":"Now, I can still substitute more because the square root of x squared plus 1,"},{"Start":"03:29.900 ","End":"03:31.610","Text":"it\u0027s just our original substitution,"},{"Start":"03:31.610 ","End":"03:35.035","Text":"is t. Here I have t"},{"Start":"03:35.035 ","End":"03:43.170","Text":"over and then I have x squared and another t, dt."},{"Start":"03:43.170 ","End":"03:46.930","Text":"Again, I\u0027ve got some cancellation,"},{"Start":"03:46.930 ","End":"03:52.880","Text":"this t with this t. But my problem now is that I have x squared,"},{"Start":"03:52.880 ","End":"03:55.430","Text":"but I don\u0027t have it in terms of t. Remember,"},{"Start":"03:55.430 ","End":"03:56.540","Text":"in the middle step here,"},{"Start":"03:56.540 ","End":"04:00.300","Text":"we had x squared was t squared minus 1."},{"Start":"04:00.500 ","End":"04:09.320","Text":"Finally, what I get is the integral of 1 over x squared,"},{"Start":"04:09.320 ","End":"04:13.150","Text":"as I said, is t squared minus 1 dt."},{"Start":"04:13.150 ","End":"04:18.905","Text":"Boy oh boy, I think this is going to be a partial fractions thing. Let\u0027s see."},{"Start":"04:18.905 ","End":"04:25.310","Text":"This was going to make it the integral of 1 over this."},{"Start":"04:25.310 ","End":"04:26.400","Text":"If I factorize it,"},{"Start":"04:26.400 ","End":"04:31.215","Text":"it is t minus 1, t plus 1."},{"Start":"04:31.215 ","End":"04:37.070","Text":"I don\u0027t want to get bogged down with partial fractions dt."},{"Start":"04:37.070 ","End":"04:45.525","Text":"Basically, what you do is you rewrite this as a over t minus 1,"},{"Start":"04:45.525 ","End":"04:49.590","Text":"plus b over t plus 1."},{"Start":"04:49.590 ","End":"04:53.270","Text":"Using partial fractions methods, we find a and b,"},{"Start":"04:53.270 ","End":"04:54.980","Text":"but I\u0027ll just tell you what a and b are,"},{"Start":"04:54.980 ","End":"04:57.260","Text":"cause I don\u0027t want to get bogged down with this."},{"Start":"04:57.260 ","End":"04:58.910","Text":"I just wrote the answer,"},{"Start":"04:58.910 ","End":"05:01.535","Text":"a is a 1/2, b is minus a 1/2."},{"Start":"05:01.535 ","End":"05:10.765","Text":"That means that I get the integral of 1/2 over t minus 1dt,"},{"Start":"05:10.765 ","End":"05:20.155","Text":"of course, plus the integral of minus a 1/2 over t plus 1dt."},{"Start":"05:20.155 ","End":"05:24.345","Text":"I\u0027ll take the 1/2 outside the brackets. You\u0027ll see."},{"Start":"05:24.345 ","End":"05:30.510","Text":"Now, the integral of 1 over t minus 1 is natural log of t minus 1."},{"Start":"05:30.510 ","End":"05:32.655","Text":"It\u0027s an absolute value."},{"Start":"05:32.655 ","End":"05:37.640","Text":"This one will be minus because of this minus natural log"},{"Start":"05:37.640 ","End":"05:42.470","Text":"of absolute value of t plus 1 and the a 1/2 I took outside,"},{"Start":"05:42.470 ","End":"05:44.710","Text":"and plus a constant."},{"Start":"05:44.710 ","End":"05:49.160","Text":"Now, we have to substitute from t back to x."},{"Start":"05:49.160 ","End":"05:55.790","Text":"As I recall, t was the square root of x squared plus 1."},{"Start":"05:55.790 ","End":"06:01.400","Text":"This is 1/2 the natural log"},{"Start":"06:01.400 ","End":"06:11.395","Text":"of absolute value of t minus 1 is square root of x squared plus 1 minus 1,"},{"Start":"06:11.395 ","End":"06:16.280","Text":"minus the natural log of absolute value"},{"Start":"06:16.280 ","End":"06:22.865","Text":"of square root of x squared plus 1 plus 1."},{"Start":"06:22.865 ","End":"06:25.834","Text":"First of all, I can drop the absolute value"},{"Start":"06:25.834 ","End":"06:31.300","Text":"because the square root of x squared plus 1 is bigger or equal to 1."},{"Start":"06:31.300 ","End":"06:34.820","Text":"Something bigger or equal to 1 minus 1 is non-negative,"},{"Start":"06:34.820 ","End":"06:35.900","Text":"so I can remove that."},{"Start":"06:35.900 ","End":"06:38.400","Text":"Similarly here, this is bigger or equal to 1,"},{"Start":"06:38.400 ","End":"06:40.685","Text":"so altogether it\u0027s bigger or equal to 2."},{"Start":"06:40.685 ","End":"06:42.815","Text":"I can drop the absolute value."},{"Start":"06:42.815 ","End":"06:44.600","Text":"Hang on. Well, didn\u0027t drop them,"},{"Start":"06:44.600 ","End":"06:46.580","Text":"replace them with brackets."},{"Start":"06:46.580 ","End":"06:51.380","Text":"Then I can use the fact that the logarithm to"},{"Start":"06:51.380 ","End":"06:56.930","Text":"any base of a minus the logarithm of b is the logarithm"},{"Start":"06:56.930 ","End":"07:02.195","Text":"of a over b. I can rewrite this now"},{"Start":"07:02.195 ","End":"07:10.090","Text":"as 1/2 the natural logarithm of a one big quotient here."},{"Start":"07:10.090 ","End":"07:16.430","Text":"This one, square root of x squared plus 1 minus 1,"},{"Start":"07:16.430 ","End":"07:22.085","Text":"over square root of x squared plus 1 plus 1."},{"Start":"07:22.085 ","End":"07:23.270","Text":"Technically, by the way,"},{"Start":"07:23.270 ","End":"07:25.460","Text":"I should have kept the plus C,"},{"Start":"07:25.460 ","End":"07:27.680","Text":"but since we\u0027re going to be doing a definite integral,"},{"Start":"07:27.680 ","End":"07:28.925","Text":"we don\u0027t need that."},{"Start":"07:28.925 ","End":"07:31.655","Text":"Hang on while I copy-paste."},{"Start":"07:31.655 ","End":"07:36.415","Text":"Copy the indefinite integral over here."},{"Start":"07:36.415 ","End":"07:41.225","Text":"Now, what I have to do is plug it in here."},{"Start":"07:41.225 ","End":"07:51.270","Text":"What I get is the limit as a goes to 0 from above of this thing."},{"Start":"07:51.270 ","End":"07:54.555","Text":"I\u0027ll write the 1/2 in front. That\u0027s the 1/2 there."},{"Start":"07:54.555 ","End":"07:57.315","Text":"All I\u0027m left with is quite messy really,"},{"Start":"07:57.315 ","End":"07:59.445","Text":"is this natural logarithm."},{"Start":"07:59.445 ","End":"08:00.890","Text":"I\u0027ll fill it in in a moment."},{"Start":"08:00.890 ","End":"08:06.860","Text":"I\u0027m going to take it between the limits of a and 1."},{"Start":"08:06.860 ","End":"08:08.690","Text":"Let me just rewrite this here."},{"Start":"08:08.690 ","End":"08:12.980","Text":"The square root of x squared plus 1 minus 1,"},{"Start":"08:12.980 ","End":"08:17.315","Text":"over the square root of x squared plus 1, plus 1."},{"Start":"08:17.315 ","End":"08:20.510","Text":"Let\u0027s see, we have to substitute 1,"},{"Start":"08:20.510 ","End":"08:23.475","Text":"then we have to substitute a, and then subtract."},{"Start":"08:23.475 ","End":"08:24.950","Text":"The part with 1,"},{"Start":"08:24.950 ","End":"08:26.420","Text":"let\u0027s see what we get."},{"Start":"08:26.420 ","End":"08:30.650","Text":"We get square root of 2 minus 1 over square root of 2 plus 1."},{"Start":"08:30.650 ","End":"08:32.240","Text":"It\u0027s just some constant."},{"Start":"08:32.240 ","End":"08:34.490","Text":"I\u0027ll just write it out. It\u0027s the best."},{"Start":"08:34.490 ","End":"08:41.360","Text":"It\u0027s 1/2 of natural log of square root of 2 minus 1,"},{"Start":"08:41.360 ","End":"08:44.285","Text":"over square root of 2 plus 1,"},{"Start":"08:44.285 ","End":"08:54.230","Text":"minus the natural log of square root of a squared plus 1 minus 1,"},{"Start":"08:54.230 ","End":"09:01.370","Text":"over the square root of a squared plus 1, plus 1."},{"Start":"09:01.370 ","End":"09:03.640","Text":"I forgot to write the limit."},{"Start":"09:03.640 ","End":"09:05.710","Text":"Forgive me. I\u0027ll just write it in here."},{"Start":"09:05.710 ","End":"09:10.930","Text":"It\u0027s the limit as A goes to 0 from above."},{"Start":"09:10.930 ","End":"09:13.075","Text":"Now, I don\u0027t need this anymore."},{"Start":"09:13.075 ","End":"09:16.255","Text":"Now, this looks worse than it actually is."},{"Start":"09:16.255 ","End":"09:18.325","Text":"Let me focus on one part of it,"},{"Start":"09:18.325 ","End":"09:23.470","Text":"on the limit as a goes to 0 plus of this bit here."},{"Start":"09:23.470 ","End":"09:27.700","Text":"If a goes to 0 plus,"},{"Start":"09:27.700 ","End":"09:34.755","Text":"then a squared plus 1 goes to 1 plus."},{"Start":"09:34.755 ","End":"09:38.430","Text":"The whole numerator goes to 1 plus minus 1,"},{"Start":"09:38.430 ","End":"09:43.655","Text":"which is 0 plus and the denominator is positive."},{"Start":"09:43.655 ","End":"09:48.650","Text":"Basically, what we get here is the natural log of 0 plus over something,"},{"Start":"09:48.650 ","End":"09:54.385","Text":"which is just 0 plus and this is known to be minus infinity."},{"Start":"09:54.385 ","End":"09:57.155","Text":"This constant here is not going to make any difference."},{"Start":"09:57.155 ","End":"09:59.555","Text":"Dividing by 2 is not going to make any difference."},{"Start":"09:59.555 ","End":"10:01.745","Text":"But we do have a minus here."},{"Start":"10:01.745 ","End":"10:06.050","Text":"Basically, this minus logarithm"},{"Start":"10:06.050 ","End":"10:11.195","Text":"tends to infinity and then the whole thing goes to infinity."},{"Start":"10:11.195 ","End":"10:14.900","Text":"The answer is that the integral diverges."},{"Start":"10:14.900 ","End":"10:18.120","Text":"That\u0027s the answer and we\u0027re done."}],"ID":4597},{"Watched":false,"Name":"Exercise 28","Duration":"4m 39s","ChapterTopicVideoID":4589,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Here we have to compute an improper integral."},{"Start":"00:02.730 ","End":"00:04.830","Text":"I\u0027ll show you later why it\u0027s improper."},{"Start":"00:04.830 ","End":"00:09.120","Text":"Because it\u0027s improper, it\u0027s relevant to ask whether it converges or diverges,"},{"Start":"00:09.120 ","End":"00:10.365","Text":"we have to decide which,"},{"Start":"00:10.365 ","End":"00:12.960","Text":"and if it converges, we can also determine its value."},{"Start":"00:12.960 ","End":"00:14.865","Text":"I\u0027ve copied it here,"},{"Start":"00:14.865 ","End":"00:17.265","Text":"and let\u0027s see why is it improper?"},{"Start":"00:17.265 ","End":"00:19.500","Text":"The integral is from 0 to 1,"},{"Start":"00:19.500 ","End":"00:25.095","Text":"and that means that we\u0027re talking about xs between 0 and 1 inclusive."},{"Start":"00:25.095 ","End":"00:27.510","Text":"But look, there\u0027s an x on the denominator,"},{"Start":"00:27.510 ","End":"00:29.205","Text":"and x, if it\u0027s 0,"},{"Start":"00:29.205 ","End":"00:31.080","Text":"will make the function undefined."},{"Start":"00:31.080 ","End":"00:32.715","Text":"When x is 0,"},{"Start":"00:32.715 ","End":"00:35.650","Text":"it\u0027s undefined, but more than that, it\u0027s also unbounded."},{"Start":"00:35.650 ","End":"00:38.025","Text":"Because when x is very small,"},{"Start":"00:38.025 ","End":"00:42.200","Text":"then 1 over x is very large and so this thing is unbounded and"},{"Start":"00:42.200 ","End":"00:46.580","Text":"undefined at 0 so it\u0027s type 2 improper integral."},{"Start":"00:46.580 ","End":"00:54.485","Text":"The way we deal with this is to replace the 0 by something that just tends to 0."},{"Start":"00:54.485 ","End":"00:56.210","Text":"I\u0027ll use the letter a,"},{"Start":"00:56.210 ","End":"01:01.165","Text":"and take the limit as a goes to 0."},{"Start":"01:01.165 ","End":"01:04.245","Text":"But not just 0, see,"},{"Start":"01:04.245 ","End":"01:06.415","Text":"if I\u0027m between 0 and 1,"},{"Start":"01:06.415 ","End":"01:11.300","Text":"then I can only go to 0 from the right or from above and everything else is the same."},{"Start":"01:11.300 ","End":"01:18.660","Text":"Upper limit is 1, 1 over x square root of x, dx."},{"Start":"01:19.060 ","End":"01:23.510","Text":"I need the indefinite integral before I can continue,"},{"Start":"01:23.510 ","End":"01:26.090","Text":"so let me do that as a side computation."},{"Start":"01:26.090 ","End":"01:32.430","Text":"What I want is the integral of 1 over x,"},{"Start":"01:32.430 ","End":"01:34.395","Text":"square root of x,"},{"Start":"01:34.395 ","End":"01:39.815","Text":"and this is equal to 1 over x to the power of 1,"},{"Start":"01:39.815 ","End":"01:42.290","Text":"x to the power of a 0.5,"},{"Start":"01:42.290 ","End":"01:48.695","Text":"which gives me 1 over x to the power of 1.5 by laws of exponents."},{"Start":"01:48.695 ","End":"01:50.750","Text":"But 1.5 is 3/2."},{"Start":"01:50.750 ","End":"01:53.760","Text":"I\u0027m just going to write that also."},{"Start":"01:53.760 ","End":"01:59.490","Text":"1.5 equals 3/2, and 1 plus 1/2 is 1.5 dx."},{"Start":"01:59.490 ","End":"02:01.550","Text":"Then because of the 1 over,"},{"Start":"02:01.550 ","End":"02:02.975","Text":"I can make it as a minus,"},{"Start":"02:02.975 ","End":"02:06.755","Text":"so it\u0027s x to the minus 3/2 dx."},{"Start":"02:06.755 ","End":"02:08.060","Text":"Some of you with experience,"},{"Start":"02:08.060 ","End":"02:12.665","Text":"will be able to get from here to here mentally."},{"Start":"02:12.665 ","End":"02:15.810","Text":"Now the integral, it\u0027s an exponent,"},{"Start":"02:15.810 ","End":"02:18.165","Text":"so we raise the exponent by 1,"},{"Start":"02:18.165 ","End":"02:22.110","Text":"that makes it x to the minus 1/2 is minus 1.5"},{"Start":"02:22.110 ","End":"02:25.980","Text":"plus 1 minus 1/2 and we divide by the new exponent,"},{"Start":"02:25.980 ","End":"02:29.850","Text":"we divide it by minus 1/2 plus a constant,"},{"Start":"02:29.850 ","End":"02:32.045","Text":"we don\u0027t need the constant here,"},{"Start":"02:32.045 ","End":"02:34.910","Text":"but I will slightly rewrite it"},{"Start":"02:34.910 ","End":"02:39.860","Text":"as take the minus 1/2 upfront and then it will become minus 2,"},{"Start":"02:39.860 ","End":"02:41.735","Text":"and the minus can come outside."},{"Start":"02:41.735 ","End":"02:47.800","Text":"It\u0027s minus the limit as a goes to 0 plus."},{"Start":"02:47.800 ","End":"02:50.475","Text":"This bit is the same,"},{"Start":"02:50.475 ","End":"02:53.265","Text":"taking the minus 1/2 is minus 2,"},{"Start":"02:53.265 ","End":"02:55.870","Text":"x to the minus 1/2,"},{"Start":"02:56.180 ","End":"03:00.040","Text":"in fact the whole minus 2 can come out front,"},{"Start":"03:00.040 ","End":"03:03.520","Text":"actually, the whole minus 2."},{"Start":"03:03.680 ","End":"03:08.535","Text":"What it is, is x to the minus 1/2,"},{"Start":"03:08.535 ","End":"03:11.360","Text":"or 1 over x to the 1/2,"},{"Start":"03:11.360 ","End":"03:18.170","Text":"and also write it that way between a and 1."},{"Start":"03:18.330 ","End":"03:27.365","Text":"What I do is I have to substitute 1.5 to substitute a and subtract."},{"Start":"03:27.365 ","End":"03:31.279","Text":"If I substitute 1 over 1 to the power of 1/2,"},{"Start":"03:31.279 ","End":"03:34.075","Text":"1 over square root of 1 is just 1."},{"Start":"03:34.075 ","End":"03:37.040","Text":"Then minus, if I put a in here,"},{"Start":"03:37.040 ","End":"03:39.680","Text":"I get 1 over a to the 1/2,"},{"Start":"03:39.680 ","End":"03:42.485","Text":"and that\u0027s the square root of a."},{"Start":"03:42.485 ","End":"03:46.355","Text":"Now, I have to take the limit as a goes to 0"},{"Start":"03:46.355 ","End":"03:50.435","Text":"from above and somewhere I lost a minus there it is."},{"Start":"03:50.435 ","End":"03:53.735","Text":"Look, when a goes to 0 plus,"},{"Start":"03:53.735 ","End":"03:56.035","Text":"then the square root of a,"},{"Start":"03:56.035 ","End":"03:58.590","Text":"also goes to 0 plus,"},{"Start":"03:58.590 ","End":"04:05.555","Text":"so 1 over square root of a goes to 1 over 0 plus which is infinity."},{"Start":"04:05.555 ","End":"04:07.805","Text":"In other words, this particular term,"},{"Start":"04:07.805 ","End":"04:11.060","Text":"the 1 over the square root of a goes to infinity."},{"Start":"04:11.060 ","End":"04:13.490","Text":"With the minuses it could be plus or"},{"Start":"04:13.490 ","End":"04:16.010","Text":"minus infinity actually comes up plus infinity in the end."},{"Start":"04:16.010 ","End":"04:17.960","Text":"But, it\u0027s not defined,"},{"Start":"04:17.960 ","End":"04:20.685","Text":"it\u0027s not a number, it diverges."},{"Start":"04:20.685 ","End":"04:23.660","Text":"Because this goes to 1 minus infinity,"},{"Start":"04:23.660 ","End":"04:25.115","Text":"which is minus infinity,"},{"Start":"04:25.115 ","End":"04:28.370","Text":"and minus 2 times minus infinity is plus infinity."},{"Start":"04:28.370 ","End":"04:31.895","Text":"Basically, this thing goes to infinity."},{"Start":"04:31.895 ","End":"04:36.740","Text":"It means that it diverges and that\u0027s what we want to know,"},{"Start":"04:36.740 ","End":"04:40.020","Text":"so we don\u0027t have to compute it anymore. We are done."}],"ID":4598},{"Watched":false,"Name":"Exercise 29","Duration":"7m 4s","ChapterTopicVideoID":4590,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.304","Text":"Here we have to find the area under the curve in the picture."},{"Start":"00:04.304 ","End":"00:06.270","Text":"I don\u0027t know why it\u0027s in 2 colors,"},{"Start":"00:06.270 ","End":"00:08.595","Text":"probably just for artistic reasons."},{"Start":"00:08.595 ","End":"00:12.960","Text":"In any event, we have to evaluate the area as an integral."},{"Start":"00:12.960 ","End":"00:15.690","Text":"The area is all above the x-axis."},{"Start":"00:15.690 ","End":"00:18.300","Text":"That\u0027s straightforward. Let\u0027s see."},{"Start":"00:18.300 ","End":"00:24.225","Text":"This is what must be the curve y equals xe to the minus x, that\u0027s here."},{"Start":"00:24.225 ","End":"00:28.035","Text":"X is bigger or equal to 0 means it\u0027s from 0 and to the right,"},{"Start":"00:28.035 ","End":"00:32.760","Text":"but there\u0027s no upper limit it goes to infinity."},{"Start":"00:32.760 ","End":"00:38.750","Text":"We get an improper integral from 0 to"},{"Start":"00:38.750 ","End":"00:45.370","Text":"infinity of xe to the minus x dx."},{"Start":"00:45.370 ","End":"00:48.515","Text":"Now, we know how to handle this improper integral."},{"Start":"00:48.515 ","End":"00:53.630","Text":"What we do is replace the infinity with something that just goes to infinity."},{"Start":"00:53.630 ","End":"00:59.100","Text":"We put the limit that says the letter b goes to infinity,"},{"Start":"00:59.100 ","End":"01:08.175","Text":"and we take the integral from 0-b of xe to the minus x dx."},{"Start":"01:08.175 ","End":"01:11.660","Text":"It\u0027s like I took some b somewhere."},{"Start":"01:11.660 ","End":"01:14.570","Text":"Let\u0027s say here this was b,"},{"Start":"01:14.570 ","End":"01:16.490","Text":"and I take this area to b,"},{"Start":"01:16.490 ","End":"01:21.710","Text":"but I keep getting b going further and further towards the infinity."},{"Start":"01:21.710 ","End":"01:23.750","Text":"As b gets larger and larger,"},{"Start":"01:23.750 ","End":"01:28.085","Text":"the area between 0 and b is Rho b,"},{"Start":"01:28.085 ","End":"01:31.970","Text":"b is going to infinity and you get closer and closer."},{"Start":"01:31.970 ","End":"01:33.935","Text":"In order to continue,"},{"Start":"01:33.935 ","End":"01:36.200","Text":"I need the indefinite integral."},{"Start":"01:36.200 ","End":"01:45.130","Text":"I want the integral of xe to the minus x dx."},{"Start":"01:45.130 ","End":"01:47.975","Text":"I\u0027m going to do it by parts."},{"Start":"01:47.975 ","End":"01:55.205","Text":"The integral of u dv equals uv minus the integral of v du."},{"Start":"01:55.205 ","End":"02:03.340","Text":"I\u0027m going to let x equals u and e to the minus x dx will be dv."},{"Start":"02:03.340 ","End":"02:08.819","Text":"What I want is u, which is x,"},{"Start":"02:08.819 ","End":"02:13.265","Text":"and then I need v. The integral of e to the minus x is"},{"Start":"02:13.265 ","End":"02:17.940","Text":"minus e to the minus x. I\u0027ll put the minus here and the e to the minus x."},{"Start":"02:17.940 ","End":"02:20.300","Text":"That\u0027s uv."},{"Start":"02:20.930 ","End":"02:27.510","Text":"Here I want v du with an integral."},{"Start":"02:27.510 ","End":"02:34.140","Text":"V already have, is minus e to the minus x. I still need du."},{"Start":"02:34.140 ","End":"02:36.030","Text":"Well, if u is x,"},{"Start":"02:36.030 ","End":"02:39.430","Text":"then du is just dx."},{"Start":"02:39.440 ","End":"02:42.540","Text":"This is going to equal"},{"Start":"02:42.540 ","End":"02:51.805","Text":"minus xe to the minus x minus and minus is plus."},{"Start":"02:51.805 ","End":"02:57.995","Text":"Basically, I\u0027m canceling this minus out and making this into a plus."},{"Start":"02:57.995 ","End":"03:06.780","Text":"The integral of e to the minus x is minus e to the minus x and plus a constant,"},{"Start":"03:06.780 ","End":"03:08.480","Text":"but I\u0027m not going to write the plus a"},{"Start":"03:08.480 ","End":"03:10.910","Text":"constant because we\u0027re going to do a definite integral."},{"Start":"03:10.910 ","End":"03:13.175","Text":"I could simplify it a bit."},{"Start":"03:13.175 ","End":"03:17.795","Text":"I could take e to the minus x outside the brackets,"},{"Start":"03:17.795 ","End":"03:19.550","Text":"even with a minus,"},{"Start":"03:19.550 ","End":"03:22.100","Text":"and I\u0027m left with x plus 1."},{"Start":"03:22.100 ","End":"03:24.155","Text":"Let\u0027s do that."},{"Start":"03:24.155 ","End":"03:26.554","Text":"There\u0027s a plus C somewhere."},{"Start":"03:26.554 ","End":"03:30.290","Text":"Back here, the graph is messing up my space,"},{"Start":"03:30.290 ","End":"03:33.145","Text":"so I\u0027ll just make longer lines."},{"Start":"03:33.145 ","End":"03:42.755","Text":"This is going to equal the limit as b goes to infinity of"},{"Start":"03:42.755 ","End":"03:50.120","Text":"minus e to the minus x times"},{"Start":"03:50.120 ","End":"03:57.610","Text":"x plus 1 between 0 and b."},{"Start":"03:59.030 ","End":"04:04.255","Text":"Let\u0027s put in the value of b."},{"Start":"04:04.255 ","End":"04:09.050","Text":"If I put in b, I get continuing over here,"},{"Start":"04:09.050 ","End":"04:15.360","Text":"we get the limit as b goes to infinity."},{"Start":"04:15.360 ","End":"04:19.955","Text":"Now, plug in b and you know what? I have a better idea."},{"Start":"04:19.955 ","End":"04:24.379","Text":"Why don\u0027t I just take this minus outside"},{"Start":"04:24.379 ","End":"04:30.320","Text":"here and then I can write a minus here and I won\u0027t get messed up with the minus."},{"Start":"04:30.320 ","End":"04:33.530","Text":"All I have to do is put in b."},{"Start":"04:33.530 ","End":"04:35.630","Text":"First of all, if I put in b,"},{"Start":"04:35.630 ","End":"04:41.885","Text":"I get e to the minus b times b plus 1."},{"Start":"04:41.885 ","End":"04:44.000","Text":"Little mark, it looks like a minus,"},{"Start":"04:44.000 ","End":"04:46.610","Text":"but it\u0027s not. That\u0027s for the b."},{"Start":"04:46.610 ","End":"04:50.210","Text":"For the 0, I need to subtract,"},{"Start":"04:50.210 ","End":"04:51.875","Text":"we get a number that was minus is not here,"},{"Start":"04:51.875 ","End":"04:57.795","Text":"e to the minus 0 plus 1."},{"Start":"04:57.795 ","End":"05:04.865","Text":"Then I close the brackets and let us see what happens when b goes to infinity."},{"Start":"05:04.865 ","End":"05:06.650","Text":"This thing is a constant."},{"Start":"05:06.650 ","End":"05:08.480","Text":"The second bit, 0 plus 1 is 1,"},{"Start":"05:08.480 ","End":"05:09.620","Text":"e to the 0 is 1."},{"Start":"05:09.620 ","End":"05:12.500","Text":"This whole thing is just equal to 1,"},{"Start":"05:12.500 ","End":"05:14.570","Text":"but what about this?"},{"Start":"05:14.570 ","End":"05:20.620","Text":"I have the limit of e to the minus b times b plus 1."},{"Start":"05:20.620 ","End":"05:23.840","Text":"That\u0027s not clear because as b goes to infinity,"},{"Start":"05:23.840 ","End":"05:25.040","Text":"this goes to infinity,"},{"Start":"05:25.040 ","End":"05:28.145","Text":"but e to the minus infinity is 0."},{"Start":"05:28.145 ","End":"05:31.175","Text":"I think it\u0027s a case of L\u0027Hopital\u0027s."},{"Start":"05:31.175 ","End":"05:34.465","Text":"Well, this exercise is something."},{"Start":"05:34.465 ","End":"05:40.325","Text":"What I\u0027m saying is I want the limit of e to the minus b,"},{"Start":"05:40.325 ","End":"05:45.210","Text":"b plus 1 as b goes to infinity."},{"Start":"05:45.210 ","End":"05:47.925","Text":"I\u0027m talking about this part here."},{"Start":"05:47.925 ","End":"05:53.429","Text":"What I can do is I can write it as the limit of a quotient."},{"Start":"05:53.429 ","End":"05:55.945","Text":"With the quotients I have a chance with L\u0027Hopital\u0027s."},{"Start":"05:55.945 ","End":"06:00.755","Text":"If I put it as b plus 1 over e to the b,"},{"Start":"06:00.755 ","End":"06:02.880","Text":"b goes to infinity."},{"Start":"06:02.880 ","End":"06:04.105","Text":"Now, I have a quotient."},{"Start":"06:04.105 ","End":"06:06.085","Text":"This goes to infinity,"},{"Start":"06:06.085 ","End":"06:08.140","Text":"this goes to infinity."},{"Start":"06:08.140 ","End":"06:13.120","Text":"By L\u0027Hopital\u0027s rule, using L\u0027Hopital\u0027s for infinity over infinity,"},{"Start":"06:13.120 ","End":"06:15.790","Text":"I can replace it by the limit of"},{"Start":"06:15.790 ","End":"06:18.490","Text":"the derivative of the top over the derivative of the bottom,"},{"Start":"06:18.490 ","End":"06:24.350","Text":"which is 1 over e to the b as b goes to infinity."},{"Start":"06:24.350 ","End":"06:31.070","Text":"This time I get 1 over infinity, which is 0."},{"Start":"06:31.470 ","End":"06:36.260","Text":"I\u0027ve computed this limit as 0."},{"Start":"06:36.260 ","End":"06:38.840","Text":"I\u0027m talking about this part here."},{"Start":"06:38.840 ","End":"06:41.810","Text":"This goes to 0, this goes to 1."},{"Start":"06:41.810 ","End":"06:44.120","Text":"We have 0 minus 1."},{"Start":"06:44.120 ","End":"06:51.425","Text":"What we end up getting is minus 0 minus 1."},{"Start":"06:51.425 ","End":"06:55.760","Text":"This is here, and this ultimately equals 1."},{"Start":"06:55.760 ","End":"06:57.680","Text":"That is the answer."},{"Start":"06:57.680 ","End":"07:04.890","Text":"It converges and the area is 1. We\u0027re done."}],"ID":4599},{"Watched":false,"Name":"Exercise 30","Duration":"5m 14s","ChapterTopicVideoID":4591,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"Here we have to find the area under the curve that\u0027s sketched below."},{"Start":"00:03.750 ","End":"00:05.550","Text":"I don\u0027t know why it\u0027s in 2 colors."},{"Start":"00:05.550 ","End":"00:08.100","Text":"It\u0027s given by this function,"},{"Start":"00:08.100 ","End":"00:13.860","Text":"y equals x squared e to the minus x cubed."},{"Start":"00:13.860 ","End":"00:17.025","Text":"We do it for x bigger or equal to 0,"},{"Start":"00:17.025 ","End":"00:20.400","Text":"0 is here and bigger equal to 0 with no upper limit,"},{"Start":"00:20.400 ","End":"00:24.685","Text":"which means that we continue all the way up to infinity."},{"Start":"00:24.685 ","End":"00:27.260","Text":"The curve is always positive because"},{"Start":"00:27.260 ","End":"00:30.035","Text":"x squared is positive and e to the something is positive."},{"Start":"00:30.035 ","End":"00:35.010","Text":"But it apparently it gets smaller and smaller but never actually gets to 0."},{"Start":"00:35.230 ","End":"00:38.300","Text":"What we do is express this as an integral,"},{"Start":"00:38.300 ","End":"00:39.995","Text":"the area below a curve,"},{"Start":"00:39.995 ","End":"00:42.005","Text":"especially when the curve is positive."},{"Start":"00:42.005 ","End":"00:51.485","Text":"We get it as an integral from 0 to infinity of this function,"},{"Start":"00:51.485 ","End":"00:57.705","Text":"x squared e to the minus x cubed dx."},{"Start":"00:57.705 ","End":"01:00.900","Text":"This is improper because of the infinity."},{"Start":"01:00.900 ","End":"01:03.710","Text":"The way we deal with that is we replace"},{"Start":"01:03.710 ","End":"01:07.010","Text":"the infinity by something that just goes to infinity."},{"Start":"01:07.010 ","End":"01:10.650","Text":"We take the limit of some variables say"},{"Start":"01:10.650 ","End":"01:15.395","Text":"b and make b go to infinity instead of the actual infinity."},{"Start":"01:15.395 ","End":"01:21.485","Text":"The rest of it is the same x squared e to the minus x cubed dx."},{"Start":"01:21.485 ","End":"01:25.910","Text":"What I\u0027m saying is we\u0027re taking it from 0 up to"},{"Start":"01:25.910 ","End":"01:31.160","Text":"b but b is constantly being pushed towards infinity."},{"Start":"01:31.160 ","End":"01:34.220","Text":"Might be here, might be here, might be here,"},{"Start":"01:34.220 ","End":"01:36.650","Text":"eventually it keeps getting more closer and closer to"},{"Start":"01:36.650 ","End":"01:40.345","Text":"infinity and then the limit will get the answer."},{"Start":"01:40.345 ","End":"01:43.190","Text":"Very well, in order to continue,"},{"Start":"01:43.190 ","End":"01:46.550","Text":"I\u0027m going to need the indefinite integral of this function."},{"Start":"01:46.550 ","End":"01:56.360","Text":"I want the integral of x squared e to the minus x cubed dx."},{"Start":"01:56.360 ","End":"01:58.550","Text":"I\u0027m going to use a substitution."},{"Start":"01:58.550 ","End":"02:04.640","Text":"I can see that the derivative of x cubed is approximately x squared rather than constant,"},{"Start":"02:04.640 ","End":"02:08.360","Text":"so I\u0027m going to take the substitution that t is equal"},{"Start":"02:08.360 ","End":"02:12.615","Text":"to might as well make it minus x cubed already."},{"Start":"02:12.615 ","End":"02:22.970","Text":"Then dt will equal minus 3x squared dx."},{"Start":"02:22.970 ","End":"02:26.305","Text":"But I don\u0027t have the minus 3,"},{"Start":"02:26.305 ","End":"02:29.795","Text":"so x squared dx,"},{"Start":"02:29.795 ","End":"02:31.280","Text":"I have x squared with a dx,"},{"Start":"02:31.280 ","End":"02:38.360","Text":"so I can say that dt over minus 3 is x squared dx."},{"Start":"02:38.360 ","End":"02:39.380","Text":"I\u0027m just tweaking it,"},{"Start":"02:39.380 ","End":"02:41.235","Text":"so it looks like this."},{"Start":"02:41.235 ","End":"02:46.140","Text":"Then now I do get the integral."},{"Start":"02:46.140 ","End":"02:52.380","Text":"The x squared dx is the minus dt/3,"},{"Start":"02:52.380 ","End":"02:57.270","Text":"and e to the minus x cubed is e^t."},{"Start":"02:57.310 ","End":"03:00.275","Text":"In short, writing it down here,"},{"Start":"03:00.275 ","End":"03:02.570","Text":"I get minus 1/3,"},{"Start":"03:02.570 ","End":"03:04.670","Text":"which I take outside the brackets."},{"Start":"03:04.670 ","End":"03:10.670","Text":"The integral of e^t, dt,"},{"Start":"03:10.670 ","End":"03:13.685","Text":"e^t is just e^t in the integral,"},{"Start":"03:13.685 ","End":"03:18.690","Text":"so it\u0027s minus 1/3 e^t plus constant,"},{"Start":"03:18.690 ","End":"03:21.160","Text":"which I\u0027m not going to put."},{"Start":"03:21.620 ","End":"03:26.325","Text":"Finally, I replace t by what it is."},{"Start":"03:26.325 ","End":"03:35.130","Text":"We get minus 1/3 e to the power of minus x cubed."},{"Start":"03:35.270 ","End":"03:40.310","Text":"Now that I have the indefinite integral, I can continue."},{"Start":"03:40.310 ","End":"03:49.585","Text":"What we get is the limit as b goes to infinity of based on the integral"},{"Start":"03:49.585 ","End":"03:53.810","Text":"minus 1/3 e to"},{"Start":"03:53.810 ","End":"04:03.779","Text":"the minus x cubed between the limits of 0 and b."},{"Start":"04:03.779 ","End":"04:08.680","Text":"This equals, I can take the minus 1/3 outside,"},{"Start":"04:08.680 ","End":"04:11.345","Text":"get the limit as b goes to infinity,"},{"Start":"04:11.345 ","End":"04:12.785","Text":"and that\u0027s what I have to do,"},{"Start":"04:12.785 ","End":"04:15.649","Text":"is plug in b, plug-in 0 and subtract,"},{"Start":"04:15.649 ","End":"04:18.305","Text":"but the minus 1/3 is no longer here."},{"Start":"04:18.305 ","End":"04:24.365","Text":"What I get is just imagine the minus 1/3 is ready in front so if x is b,"},{"Start":"04:24.365 ","End":"04:27.305","Text":"we get e to the minus b cubed."},{"Start":"04:27.305 ","End":"04:31.820","Text":"If x is 0, we get e to the minus 0 cubed."},{"Start":"04:31.820 ","End":"04:37.595","Text":"Now what happens is that when b goes to infinity,"},{"Start":"04:37.595 ","End":"04:40.100","Text":"b cubed also goes to infinity,"},{"Start":"04:40.100 ","End":"04:43.075","Text":"but minus b cubed goes to minus infinity."},{"Start":"04:43.075 ","End":"04:45.480","Text":"I get, well, let me just write that,"},{"Start":"04:45.480 ","End":"04:52.405","Text":"that will be e to the minus infinity and 0 cubed minus e^0,"},{"Start":"04:52.405 ","End":"04:54.885","Text":"which is minus 1/3."},{"Start":"04:54.885 ","End":"05:00.380","Text":"Now, e to the minus infinity is 0,"},{"Start":"05:00.380 ","End":"05:04.560","Text":"e^0 is minus 1,"},{"Start":"05:04.560 ","End":"05:06.825","Text":"so minus 1/3 times minus 1,"},{"Start":"05:06.825 ","End":"05:10.590","Text":"this is just equal to 1/3."},{"Start":"05:10.590 ","End":"05:15.010","Text":"That seems to be the answer, and I\u0027m done."}],"ID":4600},{"Watched":false,"Name":"Exercise 31","Duration":"3m 27s","ChapterTopicVideoID":4592,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"We have to find the area under the curve y equals"},{"Start":"00:02.760 ","End":"00:06.870","Text":"1 over square root of x between x equals 0 and x equals 4."},{"Start":"00:06.870 ","End":"00:10.560","Text":"I presume that this is the curve that is being sketched,"},{"Start":"00:10.560 ","End":"00:15.825","Text":"that this is indeed y equals 1 over the square root of x."},{"Start":"00:15.825 ","End":"00:23.680","Text":"Let\u0027s say that this is 0 and that this is equal to 4,"},{"Start":"00:24.470 ","End":"00:28.065","Text":"I don\u0027t know why it\u0027s in 2 colors."},{"Start":"00:28.065 ","End":"00:33.300","Text":"Anyway, this is actually an improper integral and"},{"Start":"00:33.300 ","End":"00:39.300","Text":"the reason that this is an improper integral is because we have between 0 and 4,"},{"Start":"00:39.300 ","End":"00:42.310","Text":"this means 0 and 4 inclusive."},{"Start":"00:42.310 ","End":"00:46.730","Text":"That means like 0 less than or equal to x, less than or equal to 4,"},{"Start":"00:46.730 ","End":"00:49.295","Text":"because we\u0027re going to have to do it at a definite"},{"Start":"00:49.295 ","End":"00:53.209","Text":"integral and it\u0027s always under a closed interval."},{"Start":"00:53.209 ","End":"00:56.475","Text":"The 0 is not defined there,"},{"Start":"00:56.475 ","End":"00:57.890","Text":"not only is it not defined,"},{"Start":"00:57.890 ","End":"01:01.025","Text":"it\u0027s unbounded, even look at the graph it goes to infinity,"},{"Start":"01:01.025 ","End":"01:03.440","Text":"denominator goes to 0, and so on."},{"Start":"01:03.440 ","End":"01:06.635","Text":"We\u0027ll do it as an indefinite integral,"},{"Start":"01:06.635 ","End":"01:17.010","Text":"and we\u0027ll do it as the integral from 0 to 4 of 1 over the square root of x dx."},{"Start":"01:17.010 ","End":"01:19.669","Text":"But because there\u0027s problems at 0,"},{"Start":"01:19.669 ","End":"01:23.630","Text":"and this is a type 2 improper integral the way we deal"},{"Start":"01:23.630 ","End":"01:27.500","Text":"with that is that instead of the 0 we put something that just tends to 0."},{"Start":"01:27.500 ","End":"01:35.145","Text":"I\u0027ll put a here and take the limit as a goes to 0, a to 4."},{"Start":"01:35.145 ","End":"01:37.070","Text":"Strictly speaking, this is not 0,"},{"Start":"01:37.070 ","End":"01:38.550","Text":"this should be 0 plus,"},{"Start":"01:38.550 ","End":"01:41.835","Text":"it means it goes to 0 from above."},{"Start":"01:41.835 ","End":"01:47.255","Text":"What I\u0027m really saying is we\u0027re taking some, say this is a,"},{"Start":"01:47.255 ","End":"01:52.595","Text":"we\u0027re taking it from a to 4 and then letting a get close to 0,"},{"Start":"01:52.595 ","End":"01:55.070","Text":"this area keeps getting bigger and bigger and we\u0027ll"},{"Start":"01:55.070 ","End":"01:57.740","Text":"see it might converge, it might diverge."},{"Start":"01:57.740 ","End":"02:01.110","Text":"If it converges, then we can find the area."},{"Start":"02:01.670 ","End":"02:07.660","Text":"Just let me copy the rest out as is the square root of x dx,"},{"Start":"02:07.660 ","End":"02:14.090","Text":"and this is equal to the limit as a goes to 0 plus."},{"Start":"02:14.090 ","End":"02:21.005","Text":"Now this is easy to do in our heads because if it was 1 over twice the square root of x,"},{"Start":"02:21.005 ","End":"02:27.810","Text":"if I had a 2 here then it would be square root of x but it doesn\u0027t have a 2 here."},{"Start":"02:27.810 ","End":"02:30.300","Text":"How about if I also add a 2 here,"},{"Start":"02:30.300 ","End":"02:33.380","Text":"that will cancel it out and that will be okay."},{"Start":"02:33.380 ","End":"02:35.170","Text":"Then it will give us,"},{"Start":"02:35.170 ","End":"02:40.799","Text":"so I can put this 2 here in front to the limit"},{"Start":"02:40.799 ","End":"02:43.340","Text":"and I\u0027ll get 1 over twice"},{"Start":"02:43.340 ","End":"02:47.350","Text":"root x is the square root of x when we take the indefinite integral,"},{"Start":"02:47.350 ","End":"02:53.210","Text":"and this we have to take between a and 4."},{"Start":"02:53.210 ","End":"02:57.705","Text":"What we get is twice the limit,"},{"Start":"02:57.705 ","End":"03:01.239","Text":"a goes to 0 from above,"},{"Start":"03:01.239 ","End":"03:03.380","Text":"and we get substitute 4,"},{"Start":"03:03.380 ","End":"03:05.000","Text":"we get square root of 4,"},{"Start":"03:05.000 ","End":"03:09.685","Text":"substitute a, we get square root of a."},{"Start":"03:09.685 ","End":"03:12.255","Text":"Now when a goes to 0,"},{"Start":"03:12.255 ","End":"03:14.970","Text":"square root of a so just goes to 0,"},{"Start":"03:14.970 ","End":"03:18.185","Text":"so basically we just get twice."},{"Start":"03:18.185 ","End":"03:21.680","Text":"Let\u0027s write that down. When this goes to 0 twice the square root of 4,"},{"Start":"03:21.680 ","End":"03:23.135","Text":"which is 2 times 2,"},{"Start":"03:23.135 ","End":"03:27.780","Text":"which equals 4 and that\u0027s the answer."}],"ID":4601},{"Watched":false,"Name":"Exercise 32","Duration":"8m ","ChapterTopicVideoID":4593,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"Here we have to find the area under the curve y equals sine x"},{"Start":"00:04.320 ","End":"00:08.700","Text":"over square root of cosine x between Pi over 4 and Pi over 2."},{"Start":"00:08.700 ","End":"00:10.980","Text":"I presume that this figure relates to this."},{"Start":"00:10.980 ","End":"00:20.789","Text":"This is presumably the curve y equals sine x over square root of cosine x."},{"Start":"00:20.789 ","End":"00:29.040","Text":"Presumably this is Pi over 4 and this is Pi over 2."},{"Start":"00:29.040 ","End":"00:32.220","Text":"We just like to point out that if I was working in"},{"Start":"00:32.220 ","End":"00:35.520","Text":"degrees and I sometimes like to think in degrees"},{"Start":"00:35.520 ","End":"00:42.650","Text":"Pi over 4 would be say 45 degrees and Pi over 2 is 90 degrees."},{"Start":"00:42.650 ","End":"00:44.750","Text":"It just helps me think that way."},{"Start":"00:44.750 ","End":"00:49.310","Text":"I know that sine and cosine are both positive in the first quadrant."},{"Start":"00:49.310 ","End":"00:56.210","Text":"But the thing is that cosine of 90 degrees or Pi over 2 is 0."},{"Start":"00:56.210 ","End":"01:00.920","Text":"We have a problem at Pi over 2 because we have a denominator 0."},{"Start":"01:00.920 ","End":"01:04.505","Text":"That means that this function is not defined,"},{"Start":"01:04.505 ","End":"01:06.380","Text":"but more than not defined,"},{"Start":"01:06.380 ","End":"01:08.110","Text":"sine of Pi over 2 is 1."},{"Start":"01:08.110 ","End":"01:12.040","Text":"We have 1 over positive 0, which is infinity."},{"Start":"01:12.040 ","End":"01:16.860","Text":"Indeed this thing goes all the way up to infinity."},{"Start":"01:17.020 ","End":"01:19.655","Text":"It\u0027s an improper integral,"},{"Start":"01:19.655 ","End":"01:22.770","Text":"undefined and unbounded here."},{"Start":"01:23.590 ","End":"01:26.990","Text":"Well, I\u0027m getting ahead of myself."},{"Start":"01:26.990 ","End":"01:29.660","Text":"Let me first of all write an integral because"},{"Start":"01:29.660 ","End":"01:34.310","Text":"the area under a positive curve is expressed by the integral."},{"Start":"01:34.310 ","End":"01:43.220","Text":"The integral is the integral from Pi over 4 to Pi over 2 of this thing,"},{"Start":"01:43.220 ","End":"01:52.020","Text":"sine x over square root of cosine x, dx."},{"Start":"01:52.020 ","End":"01:56.160","Text":"The problem is that Pi over 2."},{"Start":"01:56.160 ","End":"02:01.820","Text":"The way we get round that is instead of Pi over 2,"},{"Start":"02:01.820 ","End":"02:06.495","Text":"we write something that tends to Pi over 2."},{"Start":"02:06.495 ","End":"02:10.630","Text":"What we get is the limit."},{"Start":"02:10.630 ","End":"02:18.410","Text":"We replaced the Pi over 2 by letter b and let b tend to Pi over 2."},{"Start":"02:18.410 ","End":"02:21.215","Text":"Here the same Pi over 4,"},{"Start":"02:21.215 ","End":"02:29.135","Text":"everything else is the same sine x over square root of cosine x, dx."},{"Start":"02:29.135 ","End":"02:33.220","Text":"To continue, we\u0027re going to need the indefinite integral of this."},{"Start":"02:33.220 ","End":"02:39.215","Text":"The integral of sine x"},{"Start":"02:39.215 ","End":"02:44.505","Text":"over square root of cosine x,"},{"Start":"02:44.505 ","End":"02:48.180","Text":"d x. I\u0027ll make a substitution,"},{"Start":"02:48.180 ","End":"02:50.840","Text":"t equals either cosine x"},{"Start":"02:50.840 ","End":"02:53.360","Text":"or square root of cosine x. I don\u0027t know if it makes a difference."},{"Start":"02:53.360 ","End":"03:00.120","Text":"I\u0027ll try for t equals square root of cosine x. Let\u0027s see."},{"Start":"03:00.950 ","End":"03:08.525","Text":"The derivative of square root is 1 over twice the square root to first of all."},{"Start":"03:08.525 ","End":"03:13.340","Text":"Then we have to multiply by the inner derivative,"},{"Start":"03:13.340 ","End":"03:15.455","Text":"which would be minus sine x,"},{"Start":"03:15.455 ","End":"03:17.640","Text":"the derivative of cosine."},{"Start":"03:20.050 ","End":"03:24.740","Text":"Let\u0027s see. You know what I see,"},{"Start":"03:24.740 ","End":"03:27.590","Text":"if I just take the minus and the 2 out,"},{"Start":"03:27.590 ","End":"03:37.700","Text":"but write it as minus 1/2 sine x over square root of cosine x."},{"Start":"03:37.700 ","End":"03:39.620","Text":"It just stared at me."},{"Start":"03:39.620 ","End":"03:41.300","Text":"I\u0027m not saying we always do it this way,"},{"Start":"03:41.300 ","End":"03:43.565","Text":"but it just looked sine x square root again,"},{"Start":"03:43.565 ","End":"03:46.865","Text":"it looks like this. That\u0027s dt."},{"Start":"03:46.865 ","End":"03:48.740","Text":"If I just want this bit,"},{"Start":"03:48.740 ","End":"03:53.120","Text":"I could take minus 2 multiply both sides by minus 2,"},{"Start":"03:53.120 ","End":"04:02.345","Text":"and I\u0027ll get minus 2dt is equal to sine x over square root of cosine x."},{"Start":"04:02.345 ","End":"04:05.525","Text":"That\u0027s really easy now,"},{"Start":"04:05.525 ","End":"04:14.725","Text":"because then this integral continuing becomes the integral of minus 2."},{"Start":"04:14.725 ","End":"04:18.275","Text":"I\u0027ll put it outside the integral of just dt."},{"Start":"04:18.275 ","End":"04:22.385","Text":"The integral of dt is t. We get"},{"Start":"04:22.385 ","End":"04:27.050","Text":"minus 2t plus C."},{"Start":"04:27.050 ","End":"04:30.950","Text":"But we don\u0027t usually bother with the plus C when we\u0027re doing indefinite integrals."},{"Start":"04:30.950 ","End":"04:34.535","Text":"But you do have to remember to switch back from t to x."},{"Start":"04:34.535 ","End":"04:42.725","Text":"The final answer for the indefinite integral is minus 2 square root of cosine x,"},{"Start":"04:42.725 ","End":"04:47.630","Text":"optionally plus c. I\u0027ll put it for those who are sticklers for detail."},{"Start":"04:47.630 ","End":"04:50.405","Text":"I\u0027m getting back to here now"},{"Start":"04:50.405 ","End":"04:54.490","Text":"and going to continue now that I know the indefinite integral."},{"Start":"04:54.490 ","End":"05:03.310","Text":"What we get is the limit as b goes to Pi over 2."},{"Start":"05:04.130 ","End":"05:07.560","Text":"You know what, I\u0027ll put the minus 2 in front."},{"Start":"05:07.560 ","End":"05:09.360","Text":"Instead of putting a minus 2 here,"},{"Start":"05:09.360 ","End":"05:11.640","Text":"I\u0027ll put minus 2 in front."},{"Start":"05:11.640 ","End":"05:15.770","Text":"Then it will be just the square root of"},{"Start":"05:15.770 ","End":"05:23.925","Text":"cosine x between Pi over 4 and b."},{"Start":"05:23.925 ","End":"05:28.170","Text":"Continuing we get minus 2."},{"Start":"05:28.170 ","End":"05:33.515","Text":"Limit b goes to Pi over 2."},{"Start":"05:33.515 ","End":"05:37.730","Text":"Technically, I should have said actually that it goes to Pi over 2"},{"Start":"05:37.730 ","End":"05:42.810","Text":"from below because we\u0027re between Pi over 4 actually,"},{"Start":"05:42.810 ","End":"05:44.135","Text":"let me be precise."},{"Start":"05:44.135 ","End":"05:47.570","Text":"This needs a minus from the left or from below."},{"Start":"05:47.570 ","End":"05:49.490","Text":"Now Log in b."},{"Start":"05:49.490 ","End":"05:56.240","Text":"So we get the square root of cosine b and"},{"Start":"05:56.240 ","End":"06:03.260","Text":"subtract the square root of cosine Pi over 4."},{"Start":"06:03.260 ","End":"06:06.060","Text":"I need to put brackets here."},{"Start":"06:06.060 ","End":"06:08.435","Text":"Let\u0027s see, we have to do a limit."},{"Start":"06:08.435 ","End":"06:19.170","Text":"Cosine of Pi over 4 is cosine of 45 degrees is 1 over the square root of 2."},{"Start":"06:21.880 ","End":"06:26.789","Text":"I\u0027ll just leave it, it\u0027s a number for the moment."},{"Start":"06:27.830 ","End":"06:30.920","Text":"Well, I\u0027ll get to it in a moment,"},{"Start":"06:30.920 ","End":"06:34.325","Text":"I\u0027ll just write that cosine of Pi over 4."},{"Start":"06:34.325 ","End":"06:39.050","Text":"This thing here is 1 over the square root of 2."},{"Start":"06:39.050 ","End":"06:45.589","Text":"That\u0027s 1 of the elementary cosine of 45 degrees."},{"Start":"06:45.589 ","End":"06:49.280","Text":"I just remember it from the picture as it might confuse you,"},{"Start":"06:49.280 ","End":"06:54.005","Text":"ignore it, that the cosine it\u0027s 45 degrees, it\u0027s 1, 1."},{"Start":"06:54.005 ","End":"06:55.310","Text":"By Pythagoras root 2,"},{"Start":"06:55.310 ","End":"06:58.960","Text":"so cosine of this angle would be 1 over square root of 2."},{"Start":"06:58.960 ","End":"07:01.350","Text":"Ignore that if it confuses."},{"Start":"07:01.350 ","End":"07:05.374","Text":"Then b goes to Pi over 2."},{"Start":"07:05.374 ","End":"07:09.755","Text":"Doesn\u0027t matter from which direction cosine of Pi over 2 is defined."},{"Start":"07:09.755 ","End":"07:13.149","Text":"Cosine of 90 degrees is 0."},{"Start":"07:13.149 ","End":"07:18.330","Text":"We get here that this thing goes to 0."},{"Start":"07:18.330 ","End":"07:20.870","Text":"Basically if I put it all together,"},{"Start":"07:20.870 ","End":"07:30.770","Text":"I\u0027ve got a minus 2 times 0 minus the square root of 1 over the square root of 2."},{"Start":"07:30.770 ","End":"07:33.600","Text":"It\u0027s a funny expression."},{"Start":"07:33.650 ","End":"07:36.500","Text":"I could leave it as that,"},{"Start":"07:36.500 ","End":"07:39.320","Text":"but I\u0027d like to put the minus with the minus and say it\u0027s"},{"Start":"07:39.320 ","End":"07:43.820","Text":"twice plus 2 another square root of the square root."},{"Start":"07:43.820 ","End":"07:48.245","Text":"I could just say it\u0027s 1 over the fourth root of 2,"},{"Start":"07:48.245 ","End":"07:52.865","Text":"because the square root of the square root is the fourth root."},{"Start":"07:52.865 ","End":"07:54.410","Text":"Or you could just leave it like this,"},{"Start":"07:54.410 ","End":"07:56.585","Text":"or you could just put it as plus 2 times."},{"Start":"07:56.585 ","End":"08:01.470","Text":"Anyway, I\u0027m leaving my answer like this and we are done."}],"ID":4602},{"Watched":false,"Name":"Exercise 33","Duration":"8m 57s","ChapterTopicVideoID":4594,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Here we have to find the area above this curve,"},{"Start":"00:03.990 ","End":"00:06.000","Text":"which is presumably what\u0027s written here,"},{"Start":"00:06.000 ","End":"00:08.910","Text":"y equals natural log of x between 0 and 1,"},{"Start":"00:08.910 ","End":"00:12.240","Text":"so I\u0027m going to assume that this is my curve y equals"},{"Start":"00:12.240 ","End":"00:20.475","Text":"natural log of x and that this is 0 and that this is 1."},{"Start":"00:20.475 ","End":"00:24.870","Text":"When it says above, it means above up to the x-axis."},{"Start":"00:24.870 ","End":"00:29.530","Text":"I should have phrased it that way."},{"Start":"00:29.960 ","End":"00:33.390","Text":"Anyway, the diagram makes it clear."},{"Start":"00:33.390 ","End":"00:37.575","Text":"Now, normally, we have below the curve and it\u0027s an integral."},{"Start":"00:37.575 ","End":"00:39.720","Text":"But if our function is negative,"},{"Start":"00:39.720 ","End":"00:42.270","Text":"which it is, because the natural log,"},{"Start":"00:42.270 ","End":"00:45.315","Text":"if x is between 0 and 1,"},{"Start":"00:45.315 ","End":"00:47.220","Text":"then the natural log is negative,"},{"Start":"00:47.220 ","End":"00:48.795","Text":"so this is all negative."},{"Start":"00:48.795 ","End":"00:51.160","Text":"What I\u0027m saying is,"},{"Start":"00:52.250 ","End":"00:56.335","Text":"to get the negative to be positive,"},{"Start":"00:56.335 ","End":"01:08.060","Text":"we need minus the integral from 0-1 of natural log of x dx."},{"Start":"01:08.850 ","End":"01:12.350","Text":"Because if we just do it this way,"},{"Start":"01:12.530 ","End":"01:17.485","Text":"we\u0027d get a negative number because this function is all negative,"},{"Start":"01:17.485 ","End":"01:20.375","Text":"but the area has to be positive."},{"Start":"01:20.375 ","End":"01:22.360","Text":"Take the absolute value,"},{"Start":"01:22.360 ","End":"01:23.440","Text":"or we know it\u0027s negative,"},{"Start":"01:23.440 ","End":"01:24.850","Text":"so we\u0027d add a minus."},{"Start":"01:24.850 ","End":"01:26.335","Text":"So far, so good."},{"Start":"01:26.335 ","End":"01:33.270","Text":"Now, we have a little problem in the sense that although this thing goes from 0-1,"},{"Start":"01:33.270 ","End":"01:40.775","Text":"which implies that x is between 0 and 1 when x is 0,"},{"Start":"01:40.775 ","End":"01:45.365","Text":"the function is not defined because we don\u0027t have natural log of 0."},{"Start":"01:45.365 ","End":"01:48.574","Text":"Natural log is only defined for positive numbers,"},{"Start":"01:48.574 ","End":"01:53.010","Text":"and you can see that here it\u0027s not defined,"},{"Start":"01:53.010 ","End":"01:56.100","Text":"first of all, and it goes to infinity,"},{"Start":"01:56.100 ","End":"01:59.195","Text":"or rather minus infinity means it\u0027s unbounded."},{"Start":"01:59.195 ","End":"02:01.565","Text":"When it\u0027s undefined and unbounded at a point,"},{"Start":"02:01.565 ","End":"02:04.130","Text":"then it\u0027s a type 2 improper integral,"},{"Start":"02:04.130 ","End":"02:10.865","Text":"and the way we tackle it is we replace the problem point which is 0 by something else,"},{"Start":"02:10.865 ","End":"02:12.950","Text":"which is not 0 but goes to 0."},{"Start":"02:12.950 ","End":"02:19.799","Text":"I\u0027ll call it a and take the limit as a goes to 0,"},{"Start":"02:19.799 ","End":"02:21.650","Text":"but a just doesn\u0027t go to 0,"},{"Start":"02:21.650 ","End":"02:23.360","Text":"it goes to 0 from the right,"},{"Start":"02:23.360 ","End":"02:25.920","Text":"so I put a 0 plus."},{"Start":"02:26.530 ","End":"02:33.375","Text":"Say, this might be a and it\u0027s going to 0, I should use color."},{"Start":"02:33.375 ","End":"02:36.029","Text":"So a is going to 0,"},{"Start":"02:36.029 ","End":"02:40.170","Text":"but it might be somewhere here."},{"Start":"02:40.170 ","End":"02:43.715","Text":"We\u0027re basically taking this area up to"},{"Start":"02:43.715 ","End":"02:47.720","Text":"a and we\u0027re slowly letting a go to 0 and at the limit,"},{"Start":"02:47.720 ","End":"02:49.780","Text":"we\u0027ll get the total area."},{"Start":"02:49.780 ","End":"02:52.515","Text":"Let me just continue now."},{"Start":"02:52.515 ","End":"02:54.000","Text":"The rest of it is the same."},{"Start":"02:54.000 ","End":"03:00.010","Text":"Here we have a 1, here natural log of x dx."},{"Start":"03:02.990 ","End":"03:06.865","Text":"Now, the integral of natural log of x,"},{"Start":"03:06.865 ","End":"03:11.605","Text":"the indefinite integral, we\u0027ve seen before already."},{"Start":"03:11.605 ","End":"03:15.370","Text":"I remember it in a previous exercise and let me just quote it here."},{"Start":"03:15.370 ","End":"03:17.710","Text":"If you don\u0027t remember,"},{"Start":"03:17.710 ","End":"03:20.650","Text":"you can always do it by parts or look it up in integration tables,"},{"Start":"03:20.650 ","End":"03:26.020","Text":"but I\u0027ll just write it that the integral of natural log of"},{"Start":"03:26.020 ","End":"03:33.930","Text":"x dx is equal to x natural log of x minus x."},{"Start":"03:33.930 ","End":"03:36.640","Text":"Other way you could verify it is by"},{"Start":"03:36.640 ","End":"03:41.345","Text":"differentiating this and you\u0027ll see that you get natural log of x."},{"Start":"03:41.345 ","End":"03:46.565","Text":"Now, we can replace this integral by,"},{"Start":"03:46.565 ","End":"03:49.260","Text":"still have a limit,"},{"Start":"03:49.640 ","End":"03:55.880","Text":"but here we write x natural log of x minus x,"},{"Start":"03:55.880 ","End":"04:00.980","Text":"no integral, but this is going to be taken between a and 1."},{"Start":"04:00.980 ","End":"04:04.019","Text":"I\u0027m going to continue down here,"},{"Start":"04:04.360 ","End":"04:14.590","Text":"jump over it and say that this is equal to the limit as a goes to 0."},{"Start":"04:14.590 ","End":"04:20.430","Text":"First substitute 1, then substitute a, then subtract."},{"Start":"04:20.430 ","End":"04:22.320","Text":"First, I substitute 1,"},{"Start":"04:22.320 ","End":"04:32.825","Text":"1 times natural log of 1 minus 1 minus,"},{"Start":"04:32.825 ","End":"04:35.555","Text":"now, this bit\u0027s going to be in brackets."},{"Start":"04:35.555 ","End":"04:41.580","Text":"It\u0027s \"a natural log of a minus a.\""},{"Start":"04:41.950 ","End":"04:49.200","Text":"Maybe for symmetry, I\u0027ll put this in brackets also and then another bracket here."},{"Start":"04:49.780 ","End":"04:56.840","Text":"Now, this thing, 1 natural log of 1 is just 0 because natural log of 1 is 0."},{"Start":"04:56.840 ","End":"04:59.250","Text":"That\u0027s this point here."},{"Start":"04:59.480 ","End":"05:04.170","Text":"This 0 minus 1 is minus 1."},{"Start":"05:04.170 ","End":"05:06.330","Text":"This doesn\u0027t need to go inside the limit,"},{"Start":"05:06.330 ","End":"05:09.760","Text":"so we got minus 1 from this bit,"},{"Start":"05:10.220 ","End":"05:19.880","Text":"and then we can have minus the limit as a goes to 0 plus and the limit,"},{"Start":"05:19.880 ","End":"05:22.700","Text":"I just need to apply to the part that has a in it,"},{"Start":"05:22.700 ","End":"05:29.160","Text":"of a natural log of a minus a."},{"Start":"05:29.690 ","End":"05:33.100","Text":"What does happen when a goes to 0?"},{"Start":"05:33.100 ","End":"05:35.200","Text":"Well, when a goes to 0,"},{"Start":"05:35.200 ","End":"05:37.900","Text":"this goes to 0 also."},{"Start":"05:37.900 ","End":"05:40.615","Text":"But what happens to a natural log of a?"},{"Start":"05:40.615 ","End":"05:45.645","Text":"This goes to 0, but log of 0 plus is minus infinity."},{"Start":"05:45.645 ","End":"05:49.360","Text":"We have a 0 times minus infinity here."},{"Start":"05:49.360 ","End":"05:55.005","Text":"I think it\u0027s time for remembering our L\u0027Hopital."},{"Start":"05:55.005 ","End":"05:58.020","Text":"I\u0027ll just take this bit."},{"Start":"05:58.020 ","End":"05:59.760","Text":"I\u0027ll just highlight it."},{"Start":"05:59.760 ","End":"06:02.565","Text":"This limit of this bit,"},{"Start":"06:02.565 ","End":"06:05.230","Text":"I\u0027ll do this at the side."},{"Start":"06:05.880 ","End":"06:09.700","Text":"Why don\u0027t I do it in this color? What the heck?"},{"Start":"06:09.700 ","End":"06:12.815","Text":"I want to know what is the limit."},{"Start":"06:12.815 ","End":"06:16.870","Text":"Now I\u0027ll start higher up so I get more room."},{"Start":"06:19.220 ","End":"06:25.500","Text":"Let\u0027s see. What is the limit as a goes to"},{"Start":"06:25.500 ","End":"06:31.260","Text":"0 of a natural log of a?"},{"Start":"06:31.260 ","End":"06:34.685","Text":"Now, whenever we have a 0 times plus or minus infinity,"},{"Start":"06:34.685 ","End":"06:35.900","Text":"we don\u0027t want a product,"},{"Start":"06:35.900 ","End":"06:38.525","Text":"we want a quotient and then we use L\u0027Hopital."},{"Start":"06:38.525 ","End":"06:44.960","Text":"The obvious thing to do is instead of keeping the a in the numerator as a multiplier,"},{"Start":"06:44.960 ","End":"06:46.790","Text":"we can put it in the denominator."},{"Start":"06:46.790 ","End":"06:48.440","Text":"I\u0027ll show you what I mean."},{"Start":"06:48.440 ","End":"06:51.530","Text":"We can take it as the limit as a goes to 0,"},{"Start":"06:51.530 ","End":"06:53.900","Text":"natural log of a,"},{"Start":"06:53.900 ","End":"06:57.765","Text":"the a goes to the bottom as 1 over a."},{"Start":"06:57.765 ","End":"07:03.470","Text":"Now we have minus infinity over plus infinity."},{"Start":"07:03.470 ","End":"07:06.170","Text":"Because when a goes to 0, but it\u0027s slightly positive,"},{"Start":"07:06.170 ","End":"07:07.430","Text":"1 over a goes to infinity,"},{"Start":"07:07.430 ","End":"07:09.700","Text":"this goes to minus infinity."},{"Start":"07:09.700 ","End":"07:13.325","Text":"Either plus or minus infinity over plus or minus infinity,"},{"Start":"07:13.325 ","End":"07:15.395","Text":"we can use L\u0027Hopital."},{"Start":"07:15.395 ","End":"07:17.240","Text":"I\u0027m just going to write down,"},{"Start":"07:17.240 ","End":"07:22.330","Text":"I\u0027m going to use L\u0027Hopital\u0027s rule."},{"Start":"07:22.910 ","End":"07:26.330","Text":"Then we get a different limit,"},{"Start":"07:26.330 ","End":"07:31.140","Text":"which is to differentiate the top and the bottom."},{"Start":"07:31.140 ","End":"07:38.885","Text":"We\u0027ve got the limit as a goes to 0 of 1 over a,"},{"Start":"07:38.885 ","End":"07:44.130","Text":"and the derivative of this is minus 1 over a squared."},{"Start":"07:45.080 ","End":"07:47.780","Text":"This is the limit."},{"Start":"07:47.780 ","End":"07:49.520","Text":"Getting out of space here,"},{"Start":"07:49.520 ","End":"07:51.590","Text":"a goes to 0."},{"Start":"07:51.590 ","End":"07:58.240","Text":"If I compute this, it comes out to be minus a because a^2 goes into the numerator."},{"Start":"07:58.240 ","End":"08:01.875","Text":"This is equal to 0,"},{"Start":"08:01.875 ","End":"08:07.350","Text":"so that means that they actually both go to 0."},{"Start":"08:07.350 ","End":"08:10.125","Text":"This 1 goes to 0 also,"},{"Start":"08:10.125 ","End":"08:17.920","Text":"so all we\u0027re left with is the minus 1."},{"Start":"08:17.990 ","End":"08:20.290","Text":"Here I noticed something,"},{"Start":"08:20.290 ","End":"08:23.105","Text":"I forgot to carry this minus all the way through."},{"Start":"08:23.105 ","End":"08:26.330","Text":"Please forgive me, but it\u0027s no big deal."},{"Start":"08:26.330 ","End":"08:29.870","Text":"Put a minus here, a minus here,"},{"Start":"08:29.870 ","End":"08:33.470","Text":"a minus here, and therefore,"},{"Start":"08:33.470 ","End":"08:37.985","Text":"we need a minus here."},{"Start":"08:37.985 ","End":"08:44.765","Text":"Minus minus 1, and so our answer is equal to minus minus 1,"},{"Start":"08:44.765 ","End":"08:46.955","Text":"which is just 1."},{"Start":"08:46.955 ","End":"08:50.285","Text":"That is the answer for this area."},{"Start":"08:50.285 ","End":"08:56.900","Text":"The area is 1. I\u0027m done."}],"ID":4603},{"Watched":false,"Name":"Exercise 34","Duration":"12m 9s","ChapterTopicVideoID":4595,"CourseChapterTopicPlaylistID":3988,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Here we have to compute an area between the graph of this function,"},{"Start":"00:04.020 ","End":"00:05.910","Text":"the x-axis and the y-axis,"},{"Start":"00:05.910 ","End":"00:09.780","Text":"or rather the area is revolved around the x-axis and we have"},{"Start":"00:09.780 ","End":"00:13.800","Text":"to find the volume of the solid of revolution that it generates."},{"Start":"00:13.800 ","End":"00:15.420","Text":"Now if you just read it as text,"},{"Start":"00:15.420 ","End":"00:18.975","Text":"it doesn\u0027t really make much sense so I already drew a picture."},{"Start":"00:18.975 ","End":"00:21.750","Text":"First of all, I drew the graph of y equals"},{"Start":"00:21.750 ","End":"00:24.210","Text":"minus natural log of x and this is something"},{"Start":"00:24.210 ","End":"00:27.450","Text":"like this goes through the point x equals 1, y is 0."},{"Start":"00:27.450 ","End":"00:29.820","Text":"But I also drew the plus."},{"Start":"00:29.820 ","End":"00:32.770","Text":"I drew both plus and minus mirror images of each other."},{"Start":"00:32.770 ","End":"00:35.120","Text":"The reason I did this is,"},{"Start":"00:35.120 ","End":"00:39.020","Text":"first of all, if I just take the original function,"},{"Start":"00:39.020 ","End":"00:44.780","Text":"it looks like this and then it\u0027s easier to explain what I mean by between the x axis,"},{"Start":"00:44.780 ","End":"00:46.670","Text":"which is this, the y-axis,"},{"Start":"00:46.670 ","End":"00:47.960","Text":"which is this and the curve."},{"Start":"00:47.960 ","End":"00:50.780","Text":"I guess they\u0027re talking about this area"},{"Start":"00:50.780 ","End":"00:54.500","Text":"here and this is going to be revolved around the x-axis."},{"Start":"00:54.500 ","End":"00:57.190","Text":"Let me put back what I erased."},{"Start":"00:57.190 ","End":"01:03.530","Text":"What we want is to take this area here and"},{"Start":"01:03.530 ","End":"01:10.240","Text":"revolve it around the x-axis so we actually get some solid."},{"Start":"01:10.240 ","End":"01:12.200","Text":"It\u0027s hard to explain."},{"Start":"01:12.200 ","End":"01:13.820","Text":"But basically we get,"},{"Start":"01:13.820 ","End":"01:15.900","Text":"if I just draw some,"},{"Start":"01:15.950 ","End":"01:19.235","Text":"we get a pyramid on it\u0027s side."},{"Start":"01:19.235 ","End":"01:21.775","Text":"This should meet up with this."},{"Start":"01:21.775 ","End":"01:23.070","Text":"Well, you get the idea."},{"Start":"01:23.070 ","End":"01:26.180","Text":"If I take this flap and rotate it about here,"},{"Start":"01:26.180 ","End":"01:28.085","Text":"we get a solid."},{"Start":"01:28.085 ","End":"01:31.100","Text":"There are formulas for solid of revolution."},{"Start":"01:31.100 ","End":"01:32.810","Text":"This point here, by the way,"},{"Start":"01:32.810 ","End":"01:34.310","Text":"is the point x equals 1."},{"Start":"01:34.310 ","End":"01:37.085","Text":"That\u0027s where natural logarithm is 0."},{"Start":"01:37.085 ","End":"01:41.240","Text":"Ultimately translates to an integral from 0 to 1."},{"Start":"01:41.240 ","End":"01:49.580","Text":"Now, the formula for solid of revolution in general between a and b with a function f is"},{"Start":"01:49.580 ","End":"01:55.415","Text":"pi times the integral"},{"Start":"01:55.415 ","End":"02:03.785","Text":"of f of x squared dx between a and b."},{"Start":"02:03.785 ","End":"02:05.240","Text":"Other words in general,"},{"Start":"02:05.240 ","End":"02:09.770","Text":"if this was a and this was b and I take a function f of x,"},{"Start":"02:09.770 ","End":"02:13.850","Text":"this is the general formula for the solid of revolution."},{"Start":"02:13.850 ","End":"02:17.555","Text":"In our case, we will get"},{"Start":"02:17.555 ","End":"02:24.350","Text":"pi times the integral from 0 to 1 of the function,"},{"Start":"02:24.350 ","End":"02:26.980","Text":"doesn\u0027t matter if I take the plus or minus for the squared."},{"Start":"02:26.980 ","End":"02:30.990","Text":"I\u0027ll just say log x squared,"},{"Start":"02:30.990 ","End":"02:32.615","Text":"I can just put the 2 here,"},{"Start":"02:32.615 ","End":"02:35.930","Text":"dx, and that should do it."},{"Start":"02:35.930 ","End":"02:40.609","Text":"The hard part is doing the indefinite integral of the natural log squared."},{"Start":"02:40.609 ","End":"02:42.440","Text":"I have the feeling we have done this already,"},{"Start":"02:42.440 ","End":"02:49.685","Text":"but I don\u0027t remember so let me quickly do this one by parts. Let\u0027s try that."},{"Start":"02:49.685 ","End":"02:58.110","Text":"I want the indefinite integral of natural log of x squared dx."},{"Start":"02:58.110 ","End":"03:01.610","Text":"Let\u0027s try it by parts."},{"Start":"03:01.610 ","End":"03:12.660","Text":"I\u0027ll take this part as u and this part as dv and this will give us,"},{"Start":"03:12.660 ","End":"03:16.370","Text":"and the formula, I\u0027ll just write it symbolically."},{"Start":"03:16.370 ","End":"03:23.285","Text":"What I need is uv minus the integral of v du,"},{"Start":"03:23.285 ","End":"03:25.640","Text":"which comes out to be u,"},{"Start":"03:25.640 ","End":"03:30.905","Text":"is natural log squared of x. V is just x,"},{"Start":"03:30.905 ","End":"03:40.475","Text":"because if dv is dx then v is just x times x minus the integral of v,"},{"Start":"03:40.475 ","End":"03:46.955","Text":"which we said is just x and du is the derivative of this,"},{"Start":"03:46.955 ","End":"03:54.485","Text":"which is twice natural log of x times internal derivative,"},{"Start":"03:54.485 ","End":"04:00.460","Text":"1 over x and dx."},{"Start":"04:00.590 ","End":"04:04.730","Text":"I\u0027ll write the x in front then it looks clearer."},{"Start":"04:04.730 ","End":"04:10.080","Text":"X natural log squared of x minus, let\u0027s see."},{"Start":"04:10.080 ","End":"04:16.835","Text":"I can bring the 2 outside upfront and I have a cancellation, so to speak."},{"Start":"04:16.835 ","End":"04:22.130","Text":"This x with this x and so all I\u0027m left with,"},{"Start":"04:22.130 ","End":"04:28.355","Text":"in fact is the integral of natural log of x dx."},{"Start":"04:28.355 ","End":"04:32.665","Text":"This we\u0027ve done before and it\u0027s equal to,"},{"Start":"04:32.665 ","End":"04:35.930","Text":"I\u0027ll write it in a moment just copy this bit first minus twice."},{"Start":"04:35.930 ","End":"04:39.950","Text":"We did this as x natural log of x"},{"Start":"04:39.950 ","End":"04:42.620","Text":"minus x. I\u0027m not bothering with"},{"Start":"04:42.620 ","End":"04:46.495","Text":"the plus constant because it\u0027s going to be a definite integral."},{"Start":"04:46.495 ","End":"04:48.570","Text":"Just to simplify it,"},{"Start":"04:48.570 ","End":"04:50.880","Text":"that\u0027s all, just to get the brackets out."},{"Start":"04:50.880 ","End":"04:54.380","Text":"X natural log squared of x minus"},{"Start":"04:54.380 ","End":"05:00.350","Text":"2x natural log of x minus x and if you really insist on it,"},{"Start":"05:00.350 ","End":"05:02.225","Text":"I could put plus C here,"},{"Start":"05:02.225 ","End":"05:05.060","Text":"but not going to bring it over there."},{"Start":"05:05.060 ","End":"05:07.625","Text":"Now back to here."},{"Start":"05:07.625 ","End":"05:14.565","Text":"We continue and we\u0027ve got pi times all of this stuff,"},{"Start":"05:14.565 ","End":"05:18.900","Text":"x natural log squared of x."},{"Start":"05:18.900 ","End":"05:21.950","Text":"Actually I could take x outside the brackets,"},{"Start":"05:21.950 ","End":"05:24.140","Text":"it appears in all of them."},{"Start":"05:24.140 ","End":"05:31.010","Text":"It\u0027s x times natural log squared of x minus 2,"},{"Start":"05:31.010 ","End":"05:36.605","Text":"natural log of x. I almost made a bad mistake."},{"Start":"05:36.605 ","End":"05:40.415","Text":"This is a minus and a minus so this is a plus."},{"Start":"05:40.415 ","End":"05:42.980","Text":"Good, I caught it in time."},{"Start":"05:42.980 ","End":"05:46.320","Text":"This would be plus 1."},{"Start":"05:54.350 ","End":"05:56.700","Text":"Just a moment."},{"Start":"05:56.700 ","End":"05:59.210","Text":"Though we cut the 0 is outside the limit,"},{"Start":"05:59.210 ","End":"06:01.430","Text":"it\u0027s an improper integral."},{"Start":"06:01.430 ","End":"06:04.460","Text":"The natural log of 0 is not defined."},{"Start":"06:04.460 ","End":"06:07.385","Text":"It\u0027s an improper integral so what we have to do,"},{"Start":"06:07.385 ","End":"06:10.220","Text":"and I\u0027ll just erase this 0 here,"},{"Start":"06:10.220 ","End":"06:13.520","Text":"I just forgot that if it\u0027s improper,"},{"Start":"06:13.520 ","End":"06:15.125","Text":"we have to interpret it."},{"Start":"06:15.125 ","End":"06:17.105","Text":"We can\u0027t take a 0."},{"Start":"06:17.105 ","End":"06:23.240","Text":"Instead of the 0, we take a number a and let"},{"Start":"06:23.240 ","End":"06:29.970","Text":"the limit as a goes to 0 instead of 0."},{"Start":"06:29.970 ","End":"06:34.190","Text":"Of course it goes to 0 from above because we are between,"},{"Start":"06:34.190 ","End":"06:37.655","Text":"so 0 plus, we are between 0 and 1,"},{"Start":"06:37.655 ","End":"06:45.720","Text":"and there\u0027s still a pi here and limit as a goes to 0 plus, from a,"},{"Start":"06:45.720 ","End":"06:51.620","Text":"everything else is the same to 1 times natural log"},{"Start":"06:51.620 ","End":"07:00.800","Text":"squared of x dx and now I have to put a limit in front of here also."},{"Start":"07:00.800 ","End":"07:09.975","Text":"Here we are, limit as a goes to 0 from the right."},{"Start":"07:09.975 ","End":"07:19.755","Text":"This has to be taken between the limits of a and 1. All right."},{"Start":"07:19.755 ","End":"07:21.790","Text":"I\u0027m just going to substitute 1,"},{"Start":"07:21.790 ","End":"07:25.455","Text":"substitute a and subtract."},{"Start":"07:25.455 ","End":"07:27.435","Text":"Let\u0027s rewrite this part first."},{"Start":"07:27.435 ","End":"07:33.275","Text":"Limit as a goes to 0 plus of pi times."},{"Start":"07:33.275 ","End":"07:38.210","Text":"Now, if I put x equals 1,"},{"Start":"07:38.210 ","End":"07:40.790","Text":"natural log of 1 is 0."},{"Start":"07:40.790 ","End":"07:46.440","Text":"This is 0, and this is 0 and this is 1."},{"Start":"07:46.440 ","End":"07:51.255","Text":"If you think about it, we\u0027ll just get 1 if we put x equals 1."},{"Start":"07:51.255 ","End":"07:53.160","Text":"Again, this is 0, this is 0,"},{"Start":"07:53.160 ","End":"07:55.845","Text":"this is 1,1 times 1 is 1."},{"Start":"07:55.845 ","End":"07:57.720","Text":"Fine. That\u0027s the upper."},{"Start":"07:57.720 ","End":"07:59.880","Text":"Now the lower, which is a,"},{"Start":"07:59.880 ","End":"08:05.165","Text":"so I get a times natural log"},{"Start":"08:05.165 ","End":"08:15.535","Text":"squared of a minus twice log a plus 1."},{"Start":"08:15.535 ","End":"08:20.315","Text":"Let\u0027s take the limit as a goes to 0."},{"Start":"08:20.315 ","End":"08:22.410","Text":"Now here\u0027s the situation."},{"Start":"08:22.410 ","End":"08:25.350","Text":"The 1 is not a problem and the Pi is not a problem."},{"Start":"08:25.350 ","End":"08:29.745","Text":"What is the problem is the limit of this bit."},{"Start":"08:29.745 ","End":"08:35.680","Text":"I\u0027ll tell you why, because we have a 0 times infinity situation."},{"Start":"08:35.680 ","End":"08:41.590","Text":"The natural log of a goes to minus infinity."},{"Start":"08:41.590 ","End":"08:43.930","Text":"But here we have a polynomial with"},{"Start":"08:43.930 ","End":"08:47.200","Text":"natural log of a something squared minus twice something plus"},{"Start":"08:47.200 ","End":"08:50.300","Text":"1 the leading power is what counts and this"},{"Start":"08:50.300 ","End":"08:53.975","Text":"goes to actually plus infinity because it\u0027s squared."},{"Start":"08:53.975 ","End":"08:57.935","Text":"This goes to 0 so we have a 0 times infinity situation."},{"Start":"08:57.935 ","End":"09:01.730","Text":"I\u0027m just taking this bit that I\u0027ve marked as an asterisk at"},{"Start":"09:01.730 ","End":"09:05.885","Text":"the side here and I\u0027m going to use L\u0027Hopital on it."},{"Start":"09:05.885 ","End":"09:09.170","Text":"But L\u0027Hopital works on quotients, not on products."},{"Start":"09:09.170 ","End":"09:11.660","Text":"I\u0027m going to write it as the limit."},{"Start":"09:11.660 ","End":"09:13.430","Text":"I\u0027m just doing this bit."},{"Start":"09:13.430 ","End":"09:16.595","Text":"A goes to 0 plus,"},{"Start":"09:16.595 ","End":"09:18.710","Text":"how do I make it a quotient?"},{"Start":"09:18.710 ","End":"09:24.790","Text":"I\u0027m going to keep the natural log squared minus twice natural log a plus 1,"},{"Start":"09:24.790 ","End":"09:29.330","Text":"but the a, I\u0027m going to write in the denominator is 1 over a."},{"Start":"09:29.330 ","End":"09:33.470","Text":"Now this is infinity but not infinity times 0,"},{"Start":"09:33.470 ","End":"09:35.645","Text":"but infinity over infinity."},{"Start":"09:35.645 ","End":"09:42.590","Text":"Now I can say that I can use from here to here I\u0027m going to use L\u0027Hopital\u0027s rule."},{"Start":"09:42.590 ","End":"09:46.800","Text":"I\u0027ll just write his name, L\u0027Hopital, French name."},{"Start":"09:46.800 ","End":"09:50.240","Text":"We replaced the limit by a new limit,"},{"Start":"09:50.240 ","End":"09:54.710","Text":"which is obtained by differentiating top and bottom."},{"Start":"09:54.710 ","End":"09:56.855","Text":"If I differentiate the top,"},{"Start":"09:56.855 ","End":"10:02.120","Text":"I get twice natural log of"},{"Start":"10:02.120 ","End":"10:06.980","Text":"a times internal derivative 1"},{"Start":"10:06.980 ","End":"10:12.485","Text":"over a and here minus twice 1 over a,"},{"Start":"10:12.485 ","End":"10:16.010","Text":"the 1 disappears over the derivative of 1 over a,"},{"Start":"10:16.010 ","End":"10:20.190","Text":"which is minus 1 over a squared."},{"Start":"10:20.980 ","End":"10:23.870","Text":"Let\u0027s see what I can do with this."},{"Start":"10:23.870 ","End":"10:29.410","Text":"I can put a squared in the numerator and I\u0027ll get a limit."},{"Start":"10:29.410 ","End":"10:31.740","Text":"A goes to 0 plus,"},{"Start":"10:31.740 ","End":"10:33.780","Text":"let me take the 2 outside."},{"Start":"10:33.780 ","End":"10:35.210","Text":"I can do that in my head."},{"Start":"10:35.210 ","End":"10:40.280","Text":"Now this a squared over a will also be a squared over a."},{"Start":"10:40.280 ","End":"10:41.660","Text":"This will just make it a,"},{"Start":"10:41.660 ","End":"10:42.770","Text":"I\u0027ll just write down the answer."},{"Start":"10:42.770 ","End":"10:52.405","Text":"I get a natural log of a minus a and that\u0027s it."},{"Start":"10:52.405 ","End":"10:56.809","Text":"This limit. Now this thing goes to 0."},{"Start":"10:56.809 ","End":"11:00.620","Text":"When a go to 0 and I still have a natural log of a."},{"Start":"11:00.620 ","End":"11:05.030","Text":"Once again, L\u0027Hopital twice the limit"},{"Start":"11:05.030 ","End":"11:12.290","Text":"of natural log of a over 1 over a,"},{"Start":"11:12.290 ","End":"11:15.050","Text":"the same trick, a goes to 0."},{"Start":"11:15.050 ","End":"11:17.599","Text":"It\u0027s twice the limit."},{"Start":"11:17.599 ","End":"11:19.400","Text":"Differentiate top and bottom."},{"Start":"11:19.400 ","End":"11:24.335","Text":"1 over a over minus 1 over a squared."},{"Start":"11:24.335 ","End":"11:28.054","Text":"Continue here. This is equal to twice the limit."},{"Start":"11:28.054 ","End":"11:30.095","Text":"It\u0027s going to be a minus out here,"},{"Start":"11:30.095 ","End":"11:37.960","Text":"and it\u0027s going to be a squared over a as a goes to 0 and this will just equal 0."},{"Start":"11:37.960 ","End":"11:39.890","Text":"Let\u0027s see where do we stand?"},{"Start":"11:39.890 ","End":"11:41.390","Text":"We\u0027re still in the asterisk."},{"Start":"11:41.390 ","End":"11:43.910","Text":"The asterisk had 2 bits;"},{"Start":"11:43.910 ","End":"11:45.170","Text":"this bit and this bit,"},{"Start":"11:45.170 ","End":"11:47.165","Text":"this also goes to 0,"},{"Start":"11:47.165 ","End":"11:51.695","Text":"which means that this whole asterisk goes to 0."},{"Start":"11:51.695 ","End":"11:55.295","Text":"We\u0027re left with just the 1 and the pi."},{"Start":"11:55.295 ","End":"12:00.205","Text":"Ultimately, this tends to pi times 1,"},{"Start":"12:00.205 ","End":"12:06.365","Text":"which is pi and that\u0027s the answer and I hope I didn\u0027t make a mistake."},{"Start":"12:06.365 ","End":"12:09.540","Text":"I\u0027m done. This was a bit of a nasty one."}],"ID":4604}],"Thumbnail":null,"ID":3988}]

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1.1

1