Introduction to Integrals Derivative Contained
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[{"Name":"Introduction to Integrals Derivative Contained","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integrals with the `Derivative Contained`","Duration":"16m 42s","ChapterTopicVideoID":1595,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/1595.jpeg","UploadDate":"2019-11-14T07:21:06.3100000","DurationForVideoObject":"PT16M42S","Description":null,"MetaTitle":"Integrals with the `Derivative Contained`: Video + Workbook | Proprep","MetaDescription":"Integrals Derivative Contained - Introduction to Integrals Derivative Contained. 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/integrals-derivative-contained/introduction-to-integrals-derivative-contained/vid1597","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.084","Text":"In this clip, I\u0027d like to introduce you to an integration technique,"},{"Start":"00:05.084 ","End":"00:08.970","Text":"which I call the derivative contained method."},{"Start":"00:08.970 ","End":"00:12.915","Text":"Simply because I couldn\u0027t think of a better name and we have to call it something."},{"Start":"00:12.915 ","End":"00:18.510","Text":"Now this derivative contained method is a technique which we can live without."},{"Start":"00:18.510 ","End":"00:22.665","Text":"There are other techniques such as integration by substitution that would work."},{"Start":"00:22.665 ","End":"00:24.690","Text":"But when it works, when it\u0027s applicable,"},{"Start":"00:24.690 ","End":"00:28.020","Text":"it works very simply and very quickly,"},{"Start":"00:28.020 ","End":"00:30.840","Text":"and so I\u0027d like to introduce you to it."},{"Start":"00:30.840 ","End":"00:37.850","Text":"Let\u0027s, for example, recall the following formula."},{"Start":"00:37.850 ","End":"00:40.205","Text":"If I have, e to the power of something,"},{"Start":"00:40.205 ","End":"00:43.710","Text":"some function of x, and I differentiate it,"},{"Start":"00:43.710 ","End":"00:45.280","Text":"what I get is,"},{"Start":"00:45.280 ","End":"00:47.905","Text":"e to the power of f the same,"},{"Start":"00:47.905 ","End":"00:54.980","Text":"times the derivative of that something that f. Now what do I conclude from this?"},{"Start":"00:54.980 ","End":"00:59.900","Text":"I can conclude that the integral of e to"},{"Start":"00:59.900 ","End":"01:08.330","Text":"the f times f prime is equal to simply e to the f. How do I conclude this?"},{"Start":"01:08.330 ","End":"01:15.860","Text":"Well, simply the same way as say if I had the x^2 derivative is equal to 2x,"},{"Start":"01:15.860 ","End":"01:21.390","Text":"then I would say that the integral of 2x, is x^2."},{"Start":"01:21.390 ","End":"01:26.090","Text":"because differentiation and integration are the opposite,"},{"Start":"01:26.090 ","End":"01:29.250","Text":"of course I have to put a plus c here."},{"Start":"01:30.530 ","End":"01:33.785","Text":"Then the question is, what is this good for?"},{"Start":"01:33.785 ","End":"01:38.940","Text":"This is actually a useful formula and I\u0027ll show you how it could be used."},{"Start":"01:39.610 ","End":"01:42.740","Text":"Let\u0027s say I\u0027m given the following exercise."},{"Start":"01:42.740 ","End":"01:46.895","Text":"What is the integral of e to the power of"},{"Start":"01:46.895 ","End":"01:54.780","Text":"x^3 times 3 x^2 dx."},{"Start":"01:54.780 ","End":"02:02.275","Text":"In this case, what I would do is observe that if I let my f be x^3,"},{"Start":"02:02.275 ","End":"02:09.005","Text":"then what I have here is exactly f prime because the derivative of x^3 is 3 x^2,"},{"Start":"02:09.005 ","End":"02:13.775","Text":"and so this fits this template exactly."},{"Start":"02:13.775 ","End":"02:17.735","Text":"I can use the formula and say that the integral is e to the f,"},{"Start":"02:17.735 ","End":"02:22.830","Text":"which means that it\u0027s e to the power of x^3 plus constant,"},{"Start":"02:22.830 ","End":"02:24.945","Text":"and so that went very quickly."},{"Start":"02:24.945 ","End":"02:31.325","Text":"Now, it\u0027s not necessary that we have it exactly the derivative."},{"Start":"02:31.325 ","End":"02:33.875","Text":"I mean, we were lucky that we had this 3 here,"},{"Start":"02:33.875 ","End":"02:38.615","Text":"but it also works when we have something close and I\u0027ll show you what I mean."},{"Start":"02:38.615 ","End":"02:43.160","Text":"Suppose I had the integral of not this but e to the x to"},{"Start":"02:43.160 ","End":"02:49.835","Text":"the fourth times x^3 dx."},{"Start":"02:49.835 ","End":"02:54.095","Text":"Now, I would like to take my f to be,"},{"Start":"02:54.095 ","End":"02:58.295","Text":"x^4, and I notice I don\u0027t exactly have the derivative."},{"Start":"02:58.295 ","End":"03:01.895","Text":"After all, the derivative of x to the fourth is 4x^3,"},{"Start":"03:01.895 ","End":"03:03.245","Text":"and I only have x^3,"},{"Start":"03:03.245 ","End":"03:05.060","Text":"I don\u0027t have the 4."},{"Start":"03:05.060 ","End":"03:10.205","Text":"No problem, what we do is we can fix that."},{"Start":"03:10.205 ","End":"03:18.720","Text":"We write it as the integral of e to the power of x to the fourth times 4x^3,"},{"Start":"03:20.630 ","End":"03:24.110","Text":"and now I have exactly what I want because I want the"},{"Start":"03:24.110 ","End":"03:26.750","Text":"4 x^3 from the derivative of x to the fourth,"},{"Start":"03:26.750 ","End":"03:29.150","Text":"but I can\u0027t just go ahead and add a 4 there."},{"Start":"03:29.150 ","End":"03:32.360","Text":"I have to compensate and also divide by 4."},{"Start":"03:32.360 ","End":"03:37.865","Text":"4 is a constant, so I can do that outside of the integral sign and write 1 quarter here."},{"Start":"03:37.865 ","End":"03:39.680","Text":"Now what I changed,"},{"Start":"03:39.680 ","End":"03:42.785","Text":"I fixed 4/4, so I haven\u0027t changed anything."},{"Start":"03:42.785 ","End":"03:46.325","Text":"As now I do have after the integral sign,"},{"Start":"03:46.325 ","End":"03:48.409","Text":"exactly what fits this formula,"},{"Start":"03:48.409 ","End":"03:51.020","Text":"and so I can write that this equals,"},{"Start":"03:51.020 ","End":"03:52.895","Text":"the quarter has to stay here."},{"Start":"03:52.895 ","End":"03:56.075","Text":"But then I used the formula and get e to the f,"},{"Start":"03:56.075 ","End":"03:59.080","Text":"which is just e to the x to the fourth,"},{"Start":"03:59.080 ","End":"04:01.755","Text":"and plus the constant, of course."},{"Start":"04:01.755 ","End":"04:07.610","Text":"As you see, it goes very quickly and it doesn\u0027t even have to be exactly f prime here,"},{"Start":"04:07.610 ","End":"04:10.320","Text":"something close and we can fix it."},{"Start":"04:11.000 ","End":"04:15.105","Text":"We got ourselves a nice little formula,"},{"Start":"04:15.105 ","End":"04:19.370","Text":"and it\u0027s really, as nice as it is,"},{"Start":"04:19.370 ","End":"04:20.765","Text":"it\u0027s just not general."},{"Start":"04:20.765 ","End":"04:23.000","Text":"If all I could get with this formula,"},{"Start":"04:23.000 ","End":"04:25.490","Text":"I wouldn\u0027t be teaching this technique."},{"Start":"04:25.490 ","End":"04:30.395","Text":"The thing is, what I did here with the exponential function e to the power of,"},{"Start":"04:30.395 ","End":"04:35.825","Text":"I can do with many other functions and get a whole series of these neat little formulas,"},{"Start":"04:35.825 ","End":"04:39.160","Text":"and all of them have the property that there\u0027s a function in,"},{"Start":"04:39.160 ","End":"04:41.030","Text":"but the derivative is also"},{"Start":"04:41.030 ","End":"04:44.450","Text":"contained here and that\u0027s why I called it the derivative contained method."},{"Start":"04:44.450 ","End":"04:45.980","Text":"Let\u0027s start with something else,"},{"Start":"04:45.980 ","End":"04:47.345","Text":"not e to the power of."},{"Start":"04:47.345 ","End":"04:50.180","Text":"Let\u0027s go to the natural logarithm."},{"Start":"04:50.180 ","End":"04:58.965","Text":"I can write, natural log of something,"},{"Start":"04:58.965 ","End":"05:00.600","Text":"if I derive it,"},{"Start":"05:00.600 ","End":"05:03.875","Text":"I get 1 over that thing,"},{"Start":"05:03.875 ","End":"05:08.615","Text":"which is f function of x times the inner derivative,"},{"Start":"05:08.615 ","End":"05:10.205","Text":"which is f prime,"},{"Start":"05:10.205 ","End":"05:17.435","Text":"or more simply, f prime over f. From this,"},{"Start":"05:17.435 ","End":"05:20.135","Text":"using the same idea as before,"},{"Start":"05:20.135 ","End":"05:24.800","Text":"I can make an integration rule that the integral of"},{"Start":"05:24.800 ","End":"05:30.780","Text":"f prime over f is equal to the natural log,"},{"Start":"05:30.780 ","End":"05:34.300","Text":"more precisely, should be the absolute value of x,"},{"Start":"05:34.300 ","End":"05:37.310","Text":"so it\u0027ll work for negative values of f also,"},{"Start":"05:37.310 ","End":"05:40.130","Text":"and plus c. Again,"},{"Start":"05:40.130 ","End":"05:43.770","Text":"I\u0027ve got myself a little integration rule,"},{"Start":"05:44.110 ","End":"05:49.610","Text":"just like I got before with the e to the exponent."},{"Start":"05:49.610 ","End":"05:52.175","Text":"Here I have 1 for the natural logarithm,"},{"Start":"05:52.175 ","End":"05:54.770","Text":"and we\u0027ll start doing this for many other functions,"},{"Start":"05:54.770 ","End":"05:57.590","Text":"and we\u0027ve got a whole series of little rules,"},{"Start":"05:57.590 ","End":"06:03.679","Text":"and all of them, where we have the function inside together with its derivative,"},{"Start":"06:03.679 ","End":"06:06.080","Text":"and that\u0027s again why I call it the derivative contained."},{"Start":"06:06.080 ","End":"06:08.015","Text":"Now how is this useful?"},{"Start":"06:08.015 ","End":"06:10.410","Text":"I\u0027ll give an example."},{"Start":"06:10.780 ","End":"06:15.965","Text":"I\u0027m looking for an example where the numerator is the derivative of the denominator,"},{"Start":"06:15.965 ","End":"06:18.515","Text":"so I can take as an example,"},{"Start":"06:18.515 ","End":"06:26.250","Text":"the integral of 2x over x^2 plus 4."},{"Start":"06:26.250 ","End":"06:29.080","Text":"Here I see that what I have is,"},{"Start":"06:29.080 ","End":"06:31.520","Text":"if f is my denominator x^2 plus 4,"},{"Start":"06:31.520 ","End":"06:34.880","Text":"the numerator f prime is exactly 2x,"},{"Start":"06:34.880 ","End":"06:40.160","Text":"so I can use this and immediately write the answer as the natural logarithm"},{"Start":"06:40.160 ","End":"06:46.920","Text":"of x^2 plus 4 plus constant."},{"Start":"06:47.170 ","End":"06:50.150","Text":"As with the case of the exponent,"},{"Start":"06:50.150 ","End":"06:56.570","Text":"we don\u0027t have to have exactly the derivative of the denominator and the numerator,"},{"Start":"06:56.570 ","End":"06:58.145","Text":"it can be something close."},{"Start":"06:58.145 ","End":"07:02.900","Text":"For example, okay, this time I\u0027ll take the"},{"Start":"07:02.900 ","End":"07:10.810","Text":"integral of x^2 over x^3 minus 1."},{"Start":"07:10.810 ","End":"07:13.670","Text":"Once again, I want to use this formula."},{"Start":"07:13.670 ","End":"07:18.080","Text":"The denominator should be f, but unfortunately,"},{"Start":"07:18.080 ","End":"07:23.420","Text":"the derivative of x^3 minus 1 is not x^2, it\u0027s 3 x^2."},{"Start":"07:23.420 ","End":"07:27.690","Text":"As we\u0027ve seen before, these constants are not really a problem,"},{"Start":"07:27.690 ","End":"07:29.180","Text":"we can easily get around it."},{"Start":"07:29.180 ","End":"07:34.225","Text":"The way to do it as before is to write what we really want here,"},{"Start":"07:34.225 ","End":"07:41.300","Text":"which is 3 x^2, this is what\u0027s going to help me over x^3 minus 1."},{"Start":"07:41.780 ","End":"07:47.025","Text":"But I can\u0027t just go ahead and change the exercise and stick a 3 there."},{"Start":"07:47.025 ","End":"07:50.145","Text":"I have to then compensate and I have to divide by 3."},{"Start":"07:50.145 ","End":"07:51.330","Text":"Now 3 is a constant,"},{"Start":"07:51.330 ","End":"07:53.925","Text":"so it can come outside of the integral sign,"},{"Start":"07:53.925 ","End":"07:57.820","Text":"so I can put the 1/3 out here."},{"Start":"07:57.830 ","End":"08:01.800","Text":"Now this 3 and this 3 compensate for each other."},{"Start":"08:01.800 ","End":"08:07.440","Text":"But this time, I do have what\u0027s in the template of the function,"},{"Start":"08:07.440 ","End":"08:10.305","Text":"f here and its derivative on top."},{"Start":"08:10.305 ","End":"08:18.375","Text":"The answer is going to simply be 1/3 of the natural log of"},{"Start":"08:18.375 ","End":"08:27.510","Text":"absolute value of x cubed minus 1 plus c. You see that when this method works,"},{"Start":"08:27.510 ","End":"08:29.745","Text":"it works very simply and quickly."},{"Start":"08:29.745 ","End":"08:32.700","Text":"We\u0027ve already generated 2 formulas,"},{"Start":"08:32.700 ","End":"08:35.700","Text":"one from e^f and one from natural log of f,"},{"Start":"08:35.700 ","End":"08:37.770","Text":"and let\u0027s just keep going."},{"Start":"08:37.770 ","End":"08:45.780","Text":"Now this derivative contained method isn\u0027t just a single formula, it\u0027s many formulas."},{"Start":"08:45.780 ","End":"08:48.940","Text":"We started out with 1 based on e^f,"},{"Start":"08:48.940 ","End":"08:52.230","Text":"and then we had 1 based on the natural log of f,"},{"Start":"08:52.230 ","End":"08:56.755","Text":"and now let\u0027s go for 1 based on the squared function on f squared."},{"Start":"08:56.755 ","End":"09:06.045","Text":"What we have is that f squared derivative is equal to twice f,"},{"Start":"09:06.045 ","End":"09:08.895","Text":"that\u0027s something squared is twice that something,"},{"Start":"09:08.895 ","End":"09:12.880","Text":"times the internal derivative of that something."},{"Start":"09:13.070 ","End":"09:19.890","Text":"From this differentiation, we get an integral."},{"Start":"09:19.890 ","End":"09:24.270","Text":"We say that the integral of 2f times f"},{"Start":"09:24.270 ","End":"09:32.820","Text":"prime dx is equal to f squared plus a constant."},{"Start":"09:32.820 ","End":"09:37.260","Text":"If we divide both sides by 2,"},{"Start":"09:37.260 ","End":"09:42.400","Text":"let me just rewrite this with the 2 on the other side."},{"Start":"09:42.470 ","End":"09:46.980","Text":"I threw the 2 over from here and it becomes 1.5 on this side,"},{"Start":"09:46.980 ","End":"09:48.930","Text":"and now we have yet a 1/3."},{"Start":"09:48.930 ","End":"09:53.130","Text":"Nice little formula that says that the integral of"},{"Start":"09:53.130 ","End":"09:58.275","Text":"a function times its derivative is 0.5 that function squared."},{"Start":"09:58.275 ","End":"10:01.320","Text":"Once again, we\u0027ve got another little formula."},{"Start":"10:01.320 ","End":"10:06.195","Text":"This method really is a whole series of formulas and in each of them,"},{"Start":"10:06.195 ","End":"10:08.400","Text":"the common thing is that the derivative of"},{"Start":"10:08.400 ","End":"10:11.895","Text":"some function is already in there. It\u0027s contained."},{"Start":"10:11.895 ","End":"10:15.360","Text":"Let\u0027s do an example with this formula,"},{"Start":"10:15.360 ","End":"10:16.845","Text":"see how this could be useful."},{"Start":"10:16.845 ","End":"10:27.310","Text":"Let\u0027s take the integral of the natural log of x over x dx."},{"Start":"10:28.610 ","End":"10:32.535","Text":"I don\u0027t see at the moment a product,"},{"Start":"10:32.535 ","End":"10:33.945","Text":"I see a quotient."},{"Start":"10:33.945 ","End":"10:36.300","Text":"But if I rewrite this just with"},{"Start":"10:36.300 ","End":"10:44.460","Text":"slight algebraic manipulation as natural log of x times 1 over x,"},{"Start":"10:44.460 ","End":"10:46.650","Text":"something we often do,"},{"Start":"10:46.650 ","End":"10:52.185","Text":"then I can see that if this is my f,"},{"Start":"10:52.185 ","End":"10:55.470","Text":"this will simply be f prime."},{"Start":"10:55.470 ","End":"10:58.635","Text":"Because the derivative of natural log of x is 1 over x,"},{"Start":"10:58.635 ","End":"11:01.275","Text":"and so it immediately fits this format."},{"Start":"11:01.275 ","End":"11:07.740","Text":"I can write the answer as 1.5 of f squared,"},{"Start":"11:07.740 ","End":"11:13.335","Text":"which is natural log of x squared plus constant."},{"Start":"11:13.335 ","End":"11:16.215","Text":"That\u0027s immediately the answer."},{"Start":"11:16.215 ","End":"11:19.710","Text":"It works very nicely when it works."},{"Start":"11:19.710 ","End":"11:23.339","Text":"Let\u0027s continue."},{"Start":"11:23.339 ","End":"11:27.225","Text":"As I mentioned, this is not the only way to do this."},{"Start":"11:27.225 ","End":"11:32.070","Text":"The technique of integration by substitution would also work here,"},{"Start":"11:32.070 ","End":"11:35.070","Text":"but this works very quickly and nicely."},{"Start":"11:35.070 ","End":"11:37.260","Text":"I want to go on to a fourth example."},{"Start":"11:37.260 ","End":"11:39.480","Text":"We did e^f, we did natural log of f,"},{"Start":"11:39.480 ","End":"11:41.025","Text":"we just did f squared."},{"Start":"11:41.025 ","End":"11:43.905","Text":"For those of you who are studying trigonometry,"},{"Start":"11:43.905 ","End":"11:49.380","Text":"we\u0027ll also do now the cosine of"},{"Start":"11:49.380 ","End":"11:58.515","Text":"f. Those of you who don\u0027t do trigonometry can skip the remainder of this clip."},{"Start":"11:58.515 ","End":"12:03.945","Text":"If I said cosine, then strictly speaking I mean sine as my starting point."},{"Start":"12:03.945 ","End":"12:06.750","Text":"Although there will be a cosine in there, you\u0027ll see."},{"Start":"12:06.750 ","End":"12:10.620","Text":"What I mean is let\u0027s take the sine of something,"},{"Start":"12:10.620 ","End":"12:15.675","Text":"some function of x, and differentiate it and see what we get."},{"Start":"12:15.675 ","End":"12:19.065","Text":"Well, the derivative of sine is cosine."},{"Start":"12:19.065 ","End":"12:21.675","Text":"First of all, we have cosine of that something,"},{"Start":"12:21.675 ","End":"12:27.765","Text":"that f. But then because it\u0027s f need the chain rule,"},{"Start":"12:27.765 ","End":"12:32.925","Text":"we need to take the internal derivative also which is f prime."},{"Start":"12:32.925 ","End":"12:39.240","Text":"From this, I can write now that the integral of cosine,"},{"Start":"12:39.240 ","End":"12:41.910","Text":"let\u0027s just put up a bracket to make it more precise,"},{"Start":"12:41.910 ","End":"12:51.225","Text":"cosine f times f prime d_x is the sine of f,"},{"Start":"12:51.225 ","End":"12:52.650","Text":"but because it\u0027s an integral,"},{"Start":"12:52.650 ","End":"12:54.525","Text":"we put plus constant."},{"Start":"12:54.525 ","End":"12:58.905","Text":"This gives us another nice little formula."},{"Start":"12:58.905 ","End":"13:02.289","Text":"I\u0027ll show you how this can be useful."},{"Start":"13:04.250 ","End":"13:11.940","Text":"First, I\u0027ll highlight the fourth rule that we\u0027ve already gotten this way."},{"Start":"13:11.940 ","End":"13:16.590","Text":"The example I\u0027ll take will be the integral of"},{"Start":"13:16.590 ","End":"13:24.970","Text":"cosine of x squared plus 1 times 2xdx."},{"Start":"13:26.300 ","End":"13:33.255","Text":"Here, if I take my function as x squared plus 1,"},{"Start":"13:33.255 ","End":"13:38.970","Text":"I can see that here I have f prime, that\u0027s the derivative."},{"Start":"13:38.970 ","End":"13:48.570","Text":"So I fit this template and I can immediately write the answer as the sine of my f,"},{"Start":"13:48.570 ","End":"13:52.620","Text":"which is x squared plus 1 plus constant."},{"Start":"13:52.620 ","End":"13:54.750","Text":"It\u0027s that quick. As usual,"},{"Start":"13:54.750 ","End":"13:57.300","Text":"if it isn\u0027t exactly the derivative,"},{"Start":"13:57.300 ","End":"13:59.054","Text":"but up to a constant,"},{"Start":"13:59.054 ","End":"14:02.290","Text":"we can fix it like we did before."},{"Start":"14:02.630 ","End":"14:06.345","Text":"I accidentally erased the board, hang on."},{"Start":"14:06.345 ","End":"14:08.460","Text":"There, at least I put the formula back."},{"Start":"14:08.460 ","End":"14:12.990","Text":"I was about to give you the last example where it didn\u0027t come out so precise."},{"Start":"14:12.990 ","End":"14:14.760","Text":"The last example, again,"},{"Start":"14:14.760 ","End":"14:23.040","Text":"using this cosine of f formula is the integral of the cosine"},{"Start":"14:23.040 ","End":"14:32.940","Text":"of x^4 minus 1 times x cubed dx."},{"Start":"14:32.940 ","End":"14:37.830","Text":"Now here, I want to let this be my f,"},{"Start":"14:37.830 ","End":"14:40.095","Text":"but I don\u0027t have f prime here."},{"Start":"14:40.095 ","End":"14:46.320","Text":"Because if this is f, then f prime is 4x cubed and I only have x cubed."},{"Start":"14:46.320 ","End":"14:48.180","Text":"We know that\u0027s not a problem."},{"Start":"14:48.180 ","End":"14:49.230","Text":"We know how to fix this."},{"Start":"14:49.230 ","End":"14:50.760","Text":"What we do is, first of all,"},{"Start":"14:50.760 ","End":"14:52.395","Text":"write what we would like,"},{"Start":"14:52.395 ","End":"14:57.780","Text":"which is the cosine of x^4 minus 1"},{"Start":"14:57.780 ","End":"15:04.605","Text":"times 4x cubed dx."},{"Start":"15:04.605 ","End":"15:10.410","Text":"This is what we\u0027d really like because here we have f and here we would have f prime."},{"Start":"15:10.410 ","End":"15:13.080","Text":"But we can\u0027t just go ahead and multiply by 4."},{"Start":"15:13.080 ","End":"15:15.465","Text":"We have to compensate and divide by 4."},{"Start":"15:15.465 ","End":"15:16.680","Text":"Since 4 is a constant,"},{"Start":"15:16.680 ","End":"15:20.745","Text":"we can take it outside the integral and write it as 5/4 here,"},{"Start":"15:20.745 ","End":"15:25.685","Text":"and now everything\u0027s fine because the 4 and the 4 compensate."},{"Start":"15:25.685 ","End":"15:29.180","Text":"But what I do have now after the integral is"},{"Start":"15:29.180 ","End":"15:34.520","Text":"exactly this formula with f being x^4 minus 1."},{"Start":"15:34.520 ","End":"15:36.470","Text":"I can immediately write the answer."},{"Start":"15:36.470 ","End":"15:38.780","Text":"After I write the 5/4,"},{"Start":"15:38.780 ","End":"15:41.720","Text":"I can write sine of my f,"},{"Start":"15:41.720 ","End":"15:48.130","Text":"which is x^4 minus 1 plus constant, of course."},{"Start":"15:48.130 ","End":"15:53.720","Text":"This formula, which was actually the fourth one,"},{"Start":"15:53.720 ","End":"15:56.770","Text":"we got by differentiating, what was it?"},{"Start":"15:56.770 ","End":"16:00.170","Text":"We, first of all, had e^f then we had natural log of f,"},{"Start":"16:00.170 ","End":"16:01.910","Text":"and then we had f squared,"},{"Start":"16:01.910 ","End":"16:08.410","Text":"and then we had the sine of f. We can do this for any number of functions."},{"Start":"16:08.410 ","End":"16:12.260","Text":"When you get adapted you\u0027ll identify immediately"},{"Start":"16:12.260 ","End":"16:16.820","Text":"the trademark of this derivative contained method."},{"Start":"16:16.820 ","End":"16:19.760","Text":"Essentially, you notice that you have a function and you have"},{"Start":"16:19.760 ","End":"16:23.060","Text":"its derivative here and some wrapper around the function."},{"Start":"16:23.060 ","End":"16:27.065","Text":"Then usually you can use the derivative contained method"},{"Start":"16:27.065 ","End":"16:31.290","Text":"even for others more general than this."},{"Start":"16:31.290 ","End":"16:36.580","Text":"But these 4 are enough for us for now."},{"Start":"16:36.590 ","End":"16:40.480","Text":"I\u0027m done with this technique."}],"ID":1597},{"Watched":false,"Name":"Exercise 1","Duration":"3m 22s","ChapterTopicVideoID":8314,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.305","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:04.305 ","End":"00:07.170","Text":"All 3 of them are going to be using the same formula,"},{"Start":"00:07.170 ","End":"00:08.850","Text":"which I\u0027ve written over here."},{"Start":"00:08.850 ","End":"00:14.460","Text":"That is, if I have an integral where the numerator is the derivative of the denominator,"},{"Start":"00:14.460 ","End":"00:19.515","Text":"then the answer is simply the natural logarithm of the denominator."},{"Start":"00:19.515 ","End":"00:22.710","Text":"It\u0027s absolute value plus the constant."},{"Start":"00:22.710 ","End":"00:24.990","Text":"Let\u0027s see. In the first 1,"},{"Start":"00:24.990 ","End":"00:26.865","Text":"let\u0027s look at the denominator."},{"Start":"00:26.865 ","End":"00:29.730","Text":"We see that it\u0027s x squared plus 1,"},{"Start":"00:29.730 ","End":"00:32.510","Text":"and the derivative is exactly the numerator,"},{"Start":"00:32.510 ","End":"00:34.505","Text":"so it fits all the conditions."},{"Start":"00:34.505 ","End":"00:40.010","Text":"In this exercise, the integral of 2x over x squared plus 1,"},{"Start":"00:40.010 ","End":"00:46.715","Text":"I can immediately write the answer as the natural log of absolute value of"},{"Start":"00:46.715 ","End":"00:50.510","Text":"the denominator plus C. I notice that"},{"Start":"00:50.510 ","End":"00:54.660","Text":"the absolute value here is superfluous because x squared plus 1 is always positive,"},{"Start":"00:54.660 ","End":"00:56.390","Text":"but it doesn\u0027t hurt to leave it in."},{"Start":"00:56.390 ","End":"00:58.595","Text":"That was part a."},{"Start":"00:58.595 ","End":"01:03.709","Text":"Now, part b is a little bit not immediate,"},{"Start":"01:03.709 ","End":"01:07.160","Text":"need to do a little bit of manipulation because if we look at b,"},{"Start":"01:07.160 ","End":"01:09.110","Text":"and let me copy it first,"},{"Start":"01:09.110 ","End":"01:13.520","Text":"x squared over x cubed plus 1."},{"Start":"01:13.520 ","End":"01:18.395","Text":"The derivative of the denominator is in fact 3x squared,"},{"Start":"01:18.395 ","End":"01:20.540","Text":"not as we have it, x squared."},{"Start":"01:20.540 ","End":"01:23.790","Text":"All we need to do is a little bit of manipulation."},{"Start":"01:23.790 ","End":"01:30.040","Text":"If I write this as 3x squared over x cubed plus 1,"},{"Start":"01:30.040 ","End":"01:32.120","Text":"then I have exactly what I want,"},{"Start":"01:32.120 ","End":"01:34.460","Text":"but I can\u0027t just go and change the exercise."},{"Start":"01:34.460 ","End":"01:40.850","Text":"If I also put 1/3 in front of it and the constant can go in and out of the integral sign,"},{"Start":"01:40.850 ","End":"01:47.180","Text":"then basically, what I\u0027m saying is that I\u0027ve compensated this 3 here with this 3 here,"},{"Start":"01:47.180 ","End":"01:48.935","Text":"makes it all okay."},{"Start":"01:48.935 ","End":"01:55.250","Text":"Now, I can just leave the 1/3 here and say that this is equal"},{"Start":"01:55.250 ","End":"02:03.320","Text":"to 1/3 of the natural log of x cubed plus 1,"},{"Start":"02:03.320 ","End":"02:06.175","Text":"in absolute value, plus C,"},{"Start":"02:06.175 ","End":"02:09.315","Text":"and that\u0027s it for part b."},{"Start":"02:09.315 ","End":"02:13.910","Text":"Now part c, also going to have to do a little bit of work."},{"Start":"02:13.910 ","End":"02:15.905","Text":"Let me first copy it."},{"Start":"02:15.905 ","End":"02:21.755","Text":"On the denominator, I have x squared plus 4x plus 1,"},{"Start":"02:21.755 ","End":"02:23.269","Text":"and on the numerator,"},{"Start":"02:23.269 ","End":"02:25.385","Text":"I have x plus 2."},{"Start":"02:25.385 ","End":"02:29.975","Text":"What I notice is that if I take the derivative of this,"},{"Start":"02:29.975 ","End":"02:32.750","Text":"what I get is 2x plus 4."},{"Start":"02:32.750 ","End":"02:35.105","Text":"That\u0027s what I need on the numerator,"},{"Start":"02:35.105 ","End":"02:37.090","Text":"but I have x plus 2."},{"Start":"02:37.090 ","End":"02:39.155","Text":"If I just double it,"},{"Start":"02:39.155 ","End":"02:40.940","Text":"I will get exactly what I want."},{"Start":"02:40.940 ","End":"02:44.105","Text":"If I multiply top and bottom by 2,"},{"Start":"02:44.105 ","End":"02:46.235","Text":"and just like I did before with the 1/3,"},{"Start":"02:46.235 ","End":"02:48.470","Text":"I\u0027ll put 1/2 out here,"},{"Start":"02:48.470 ","End":"02:51.055","Text":"but I\u0027ll multiply this by 2,"},{"Start":"02:51.055 ","End":"02:53.250","Text":"then that will be fine."},{"Start":"02:53.250 ","End":"02:58.610","Text":"Now I can definitely say that the numerator is the derivative of the denominator."},{"Start":"02:58.610 ","End":"03:00.125","Text":"This is 2x plus 4,"},{"Start":"03:00.125 ","End":"03:02.455","Text":"and the derivative of this is 2x plus 4."},{"Start":"03:02.455 ","End":"03:09.440","Text":"Now I can go ahead and use this formula and say that the answer is the natural log of"},{"Start":"03:09.440 ","End":"03:17.535","Text":"absolute value of x squared plus 4x plus 1 plus constant."},{"Start":"03:17.535 ","End":"03:23.400","Text":"That\u0027s it. All 3 of them solved with the help of this nice formula."}],"ID":8485},{"Watched":false,"Name":"Exercise 2","Duration":"3m 54s","ChapterTopicVideoID":8315,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to compute the following integrals,"},{"Start":"00:04.080 ","End":"00:07.140","Text":"and I\u0027m going to use the same formula in all 3."},{"Start":"00:07.140 ","End":"00:10.440","Text":"There\u0027s this nice formula that says that if I have the integral of"},{"Start":"00:10.440 ","End":"00:14.385","Text":"a fraction where the top is exactly the derivative of the bottom,"},{"Start":"00:14.385 ","End":"00:20.865","Text":"then the answer is natural log of absolute value of the bottom, the denominator."},{"Start":"00:20.865 ","End":"00:23.310","Text":"Let\u0027s start with the first 1,"},{"Start":"00:23.310 ","End":"00:27.945","Text":"which is the integral of cotangent x."},{"Start":"00:27.945 ","End":"00:29.580","Text":"Already, we have a problem,"},{"Start":"00:29.580 ","End":"00:31.005","Text":"we don\u0027t have a fraction here,"},{"Start":"00:31.005 ","End":"00:32.715","Text":"and then we remember, \"Ah, yes,"},{"Start":"00:32.715 ","End":"00:36.360","Text":"the cotangent is exactly the cosine over the"},{"Start":"00:36.360 ","End":"00:41.070","Text":"sine,\" so we have the integral of cosine x over sine x,"},{"Start":"00:41.070 ","End":"00:43.515","Text":"and now we do have a fraction."},{"Start":"00:43.515 ","End":"00:48.670","Text":"We see that sine x on the bottom and its derivative is exactly cosine x,"},{"Start":"00:48.670 ","End":"00:53.080","Text":"so it perfectly fits this formula with f of x being sine x,"},{"Start":"00:53.080 ","End":"00:55.595","Text":"and so we get as an answer,"},{"Start":"00:55.595 ","End":"01:04.865","Text":"the natural log of sine x and absolute value plus C. That was straightforward enough."},{"Start":"01:04.865 ","End":"01:06.530","Text":"Now, the second 1,"},{"Start":"01:06.530 ","End":"01:09.560","Text":"that\u0027s the integral of the tangent x."},{"Start":"01:09.560 ","End":"01:11.510","Text":"Once again, we don\u0027t see a fraction,"},{"Start":"01:11.510 ","End":"01:17.660","Text":"but then we remember that tangent x is simply sine x over cosine x."},{"Start":"01:17.660 ","End":"01:22.475","Text":"But this time, there\u0027s a small problem because the cosine of"},{"Start":"01:22.475 ","End":"01:28.515","Text":"x derivative is actually minus sine x,"},{"Start":"01:28.515 ","End":"01:30.440","Text":"and we have sine x."},{"Start":"01:30.440 ","End":"01:32.000","Text":"We\u0027ve seen this thing before."},{"Start":"01:32.000 ","End":"01:33.260","Text":"This is not a problem."},{"Start":"01:33.260 ","End":"01:36.920","Text":"All I have to do is put an extra minus in here and to compensate,"},{"Start":"01:36.920 ","End":"01:38.480","Text":"another minus out there,"},{"Start":"01:38.480 ","End":"01:48.155","Text":"so what I\u0027ll get will be minus the integral of minus sine x over cosine x."},{"Start":"01:48.155 ","End":"01:51.035","Text":"Now, this fits exactly this form,"},{"Start":"01:51.035 ","End":"01:52.520","Text":"but we have a minus in front,"},{"Start":"01:52.520 ","End":"02:01.355","Text":"so we have a minus the natural log of absolute value of cosine x and, of course,"},{"Start":"02:01.355 ","End":"02:05.835","Text":"plus C. The third 1, the integral,"},{"Start":"02:05.835 ","End":"02:10.805","Text":"sine x minus cosine x over"},{"Start":"02:10.805 ","End":"02:17.080","Text":"sine x plus cosine x dx."},{"Start":"02:17.080 ","End":"02:19.335","Text":"Here, I have a fraction already,"},{"Start":"02:19.335 ","End":"02:21.410","Text":"so let me check what is the derivative of"},{"Start":"02:21.410 ","End":"02:25.445","Text":"the denominator and see if I get the numerator or something close."},{"Start":"02:25.445 ","End":"02:32.815","Text":"Let\u0027s check what is the derivative of sine x plus cosine x."},{"Start":"02:32.815 ","End":"02:36.790","Text":"The derivative of sine x is cosine x,"},{"Start":"02:36.790 ","End":"02:42.230","Text":"and the derivative of cosine x is minus sine x,"},{"Start":"02:42.230 ","End":"02:44.690","Text":"but this is not what I have on the numerator here."},{"Start":"02:44.690 ","End":"02:46.430","Text":"Here, I have sine minus cosine,"},{"Start":"02:46.430 ","End":"02:48.590","Text":"and here I have cosine minus sine,"},{"Start":"02:48.590 ","End":"02:52.580","Text":"but notice that if we have a difference and we want to reverse the order,"},{"Start":"02:52.580 ","End":"02:54.370","Text":"we can just put a minus in front,"},{"Start":"02:54.370 ","End":"03:00.520","Text":"and this is actually the negative of sine x minus cosine x,"},{"Start":"03:00.520 ","End":"03:05.915","Text":"so we\u0027re in a situation very similar as in the second 1 with the tangent."},{"Start":"03:05.915 ","End":"03:08.240","Text":"What we\u0027ll have to do is the same trick again."},{"Start":"03:08.240 ","End":"03:11.500","Text":"We just have to write this as minus,"},{"Start":"03:11.500 ","End":"03:14.090","Text":"and then I can reverse the order from this."},{"Start":"03:14.090 ","End":"03:19.940","Text":"I\u0027ve got now the cosine of x minus sine x"},{"Start":"03:19.940 ","End":"03:26.600","Text":"over the same sine x plus cosine x dx."},{"Start":"03:26.600 ","End":"03:30.590","Text":"This time, I have what is written here because the derivative of"},{"Start":"03:30.590 ","End":"03:34.625","Text":"the denominator is the numerator now to which the sine is cosine,"},{"Start":"03:34.625 ","End":"03:35.750","Text":"cosine is minus sine,"},{"Start":"03:35.750 ","End":"03:37.220","Text":"so I can use the formula,"},{"Start":"03:37.220 ","End":"03:39.395","Text":"but I got to keep this minus here,"},{"Start":"03:39.395 ","End":"03:43.940","Text":"and I have the natural logarithm of the absolute value of the denominator,"},{"Start":"03:43.940 ","End":"03:48.920","Text":"sine x plus cosine x plus"},{"Start":"03:48.920 ","End":"03:55.199","Text":"C. That\u0027s the answer to the third 1 and finished with the exercise."}],"ID":8486},{"Watched":false,"Name":"Exercise 3","Duration":"4m 58s","ChapterTopicVideoID":8316,"CourseChapterTopicPlaylistID":3989,"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":"In this exercise, we have to compute the following 3 integrals."},{"Start":"00:04.650 ","End":"00:06.300","Text":"In all of these cases,"},{"Start":"00:06.300 ","End":"00:10.709","Text":"I\u0027m going to use the same formula which says if I have an integral"},{"Start":"00:10.709 ","End":"00:15.855","Text":"of a fraction where the top is exactly the derivative of the bottom,"},{"Start":"00:15.855 ","End":"00:22.290","Text":"then the answer simply the natural logarithm of the absolute value of the bottom."},{"Start":"00:22.290 ","End":"00:23.984","Text":"Let\u0027s begin with the first,"},{"Start":"00:23.984 ","End":"00:32.055","Text":"which is the integral e^x plus 2 over e^x plus 1."},{"Start":"00:32.055 ","End":"00:37.250","Text":"Now, I see a fraction and I like to see what is the derivative of the denominator."},{"Start":"00:37.250 ","End":"00:40.790","Text":"Mentally, we can easily see that this is just going to be"},{"Start":"00:40.790 ","End":"00:44.570","Text":"e^x because the constant is 0, we differentiate."},{"Start":"00:44.570 ","End":"00:47.195","Text":"But what I have on the top is not e^x,"},{"Start":"00:47.195 ","End":"00:49.920","Text":"it\u0027s something close e^x plus 2."},{"Start":"00:49.920 ","End":"00:52.460","Text":"Let\u0027s see if we can do a bit of algebra."},{"Start":"00:52.460 ","End":"00:54.020","Text":"Some other trick here."},{"Start":"00:54.020 ","End":"00:57.350","Text":"What I\u0027m going to do is use the laws of"},{"Start":"00:57.350 ","End":"01:04.970","Text":"exponents a^b plus c equals a^b times a^c."},{"Start":"01:04.970 ","End":"01:14.165","Text":"What we get is e^x times e squared over e^x plus 1 dx."},{"Start":"01:14.165 ","End":"01:17.990","Text":"Now, this e squared is a constant,"},{"Start":"01:17.990 ","End":"01:19.520","Text":"it\u0027s just a number."},{"Start":"01:19.520 ","End":"01:23.675","Text":"If so, I can take it out in front of the integral sign."},{"Start":"01:23.675 ","End":"01:31.480","Text":"E squared times the integral of e^x over e^x plus 1."},{"Start":"01:31.480 ","End":"01:35.090","Text":"Now, I\u0027m in good shape because in this integral, as we saw,"},{"Start":"01:35.090 ","End":"01:39.110","Text":"the derivative of the denominator is exactly e^x,"},{"Start":"01:39.110 ","End":"01:40.415","Text":"which is the numerator."},{"Start":"01:40.415 ","End":"01:42.620","Text":"Now, I can use this formula."},{"Start":"01:42.620 ","End":"01:47.870","Text":"The answer will be e squared times the natural log of"},{"Start":"01:47.870 ","End":"01:55.325","Text":"absolute value of e^x plus 1 plus c. That\u0027s it for the first 1."},{"Start":"01:55.325 ","End":"01:58.055","Text":"Now on to the second."},{"Start":"01:58.055 ","End":"02:01.085","Text":"I see that we do have a fraction,"},{"Start":"02:01.085 ","End":"02:04.760","Text":"but there\u0027s no way I can use this formulas is,"},{"Start":"02:04.760 ","End":"02:08.615","Text":"I mean, the derivative of the denominator is nowhere near 1."},{"Start":"02:08.615 ","End":"02:12.920","Text":"Just thinking a bit and I need some trick or some algebraic manipulation."},{"Start":"02:12.920 ","End":"02:14.240","Text":"I see after awhile,"},{"Start":"02:14.240 ","End":"02:17.300","Text":"that what I can do is rewrite it algebraically"},{"Start":"02:17.300 ","End":"02:22.325","Text":"equivalent if I write it as 1 over x over natural log of x."},{"Start":"02:22.325 ","End":"02:24.970","Text":"Just basic algebra shows this is the same,"},{"Start":"02:24.970 ","End":"02:28.535","Text":"or if you like, I can divide top and bottom by x."},{"Start":"02:28.535 ","End":"02:31.790","Text":"It\u0027s a huge advantage because now if I look at"},{"Start":"02:31.790 ","End":"02:35.255","Text":"the derivative of the denominator, natural log of x,"},{"Start":"02:35.255 ","End":"02:44.840","Text":"it is 1 over x. I can use this formula now and just get natural log of the denominator,"},{"Start":"02:44.840 ","End":"02:50.690","Text":"which is natural log of x plus c. It all turned out very simply."},{"Start":"02:50.690 ","End":"02:54.280","Text":"But you first have to see this little trick."},{"Start":"02:54.280 ","End":"02:56.760","Text":"Now that you\u0027ve seen it, if it comes up again,"},{"Start":"02:56.760 ","End":"02:59.005","Text":"you can play around algebraically."},{"Start":"02:59.005 ","End":"03:01.180","Text":"Sometimes when you get the thing that you want."},{"Start":"03:01.180 ","End":"03:04.205","Text":"Let\u0027s get on to part c now."},{"Start":"03:04.205 ","End":"03:07.320","Text":"I\u0027ve copied part c over here."},{"Start":"03:07.320 ","End":"03:14.320","Text":"Once again, I look at this form and see if I have the derivative of the denominator and"},{"Start":"03:14.320 ","End":"03:18.170","Text":"the numerator and that\u0027s no way because the derivative of this"},{"Start":"03:18.170 ","End":"03:22.680","Text":"is 2x plus 4 and this is certainly not 2x plus 4."},{"Start":"03:22.680 ","End":"03:28.150","Text":"Once again, I\u0027m going to need some trick and they\u0027ll just tell you what it is."},{"Start":"03:28.150 ","End":"03:30.610","Text":"I see I have x squared plus 4x,"},{"Start":"03:30.610 ","End":"03:33.625","Text":"and here I have also x squared plus something else."},{"Start":"03:33.625 ","End":"03:37.135","Text":"But if I break the numerator up into 2 bits,"},{"Start":"03:37.135 ","End":"03:44.695","Text":"if I take the numerator and take out x squared plus 4x plus something else,"},{"Start":"03:44.695 ","End":"03:48.035","Text":"then I might be able to divide and get 1."},{"Start":"03:48.035 ","End":"03:50.805","Text":"If I look at this, it\u0027s x squared plus 4x."},{"Start":"03:50.805 ","End":"03:54.969","Text":"All I have to do to complete it is to add another 2x,"},{"Start":"03:54.969 ","End":"03:57.880","Text":"that makes it 6x and the 4."},{"Start":"03:57.880 ","End":"04:04.005","Text":"But I\u0027ll think of it as this bit plus this bit over x squared plus 4x."},{"Start":"04:04.005 ","End":"04:05.895","Text":"Now, how does this help me?"},{"Start":"04:05.895 ","End":"04:10.080","Text":"Well, this over this will just be 1 constant."},{"Start":"04:10.080 ","End":"04:14.740","Text":"Now, I will have the derivative of the denominator in the numerator."},{"Start":"04:14.740 ","End":"04:16.480","Text":"We just said it was 2x plus 4."},{"Start":"04:16.480 ","End":"04:19.750","Text":"Let me rewrite this as the sum of 2 integrals."},{"Start":"04:19.750 ","End":"04:26.325","Text":"We get the integral of this over this is 1 dx and this"},{"Start":"04:26.325 ","End":"04:34.240","Text":"1 is 2x plus 4 over x squared plus 4x dx."},{"Start":"04:34.240 ","End":"04:37.069","Text":"We get here is a constant."},{"Start":"04:37.069 ","End":"04:38.990","Text":"The constant is a constant times x,"},{"Start":"04:38.990 ","End":"04:42.860","Text":"so it\u0027s 1x or just x plus."},{"Start":"04:42.860 ","End":"04:46.025","Text":"Now, this fits exactly the formula."},{"Start":"04:46.025 ","End":"04:55.235","Text":"We have the natural logarithm of absolute value of x squared plus 4x plus a constant."},{"Start":"04:55.235 ","End":"04:59.610","Text":"That finishes c and we\u0027re done."}],"ID":8487},{"Watched":false,"Name":"Exercise 4","Duration":"4m 2s","ChapterTopicVideoID":8317,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.780 ","End":"00:06.390","Text":"Actually, I\u0027m going to need 2 formulas."},{"Start":"00:06.390 ","End":"00:12.930","Text":"This old familiar formula I\u0027ll use in Part c and this one I\u0027ll use in parts a and b."},{"Start":"00:12.930 ","End":"00:15.990","Text":"This one says that if I have the integral"},{"Start":"00:15.990 ","End":"00:19.155","Text":"of e to the power of something, some function of x,"},{"Start":"00:19.155 ","End":"00:21.915","Text":"but I also have the derivative alongside it,"},{"Start":"00:21.915 ","End":"00:27.600","Text":"then the answer is just e to the function of x without this bit."},{"Start":"00:27.600 ","End":"00:31.140","Text":"In other words, if you find that you have e to"},{"Start":"00:31.140 ","End":"00:35.130","Text":"the something but you have the derivative of f alongside,"},{"Start":"00:35.130 ","End":"00:39.690","Text":"then the answer is simply just this e to the f. In Part a,"},{"Start":"00:39.690 ","End":"00:42.190","Text":"which is the simplest,"},{"Start":"00:42.190 ","End":"00:47.510","Text":"I do have this case exactly because if I look at the function x squared,"},{"Start":"00:47.510 ","End":"00:49.595","Text":"the x squared here,"},{"Start":"00:49.595 ","End":"00:52.040","Text":"the derivative is exactly 2x,"},{"Start":"00:52.040 ","End":"00:59.245","Text":"so I do have the case of e to the power of a function with the derivative alongside."},{"Start":"00:59.245 ","End":"01:06.035","Text":"The answer according to this is just this e to the x squared plus the constant."},{"Start":"01:06.035 ","End":"01:14.075","Text":"In Part b, we\u0027re not as lucky because if I want to look at this as e to the f,"},{"Start":"01:14.075 ","End":"01:15.560","Text":"just like I did before,"},{"Start":"01:15.560 ","End":"01:18.140","Text":"that e to the power of something,"},{"Start":"01:18.140 ","End":"01:22.000","Text":"then I don\u0027t exactly have the derivative of x cubed here"},{"Start":"01:22.000 ","End":"01:26.465","Text":"because the derivative of x cubed is 3x squared and I only have 1x squared."},{"Start":"01:26.465 ","End":"01:29.930","Text":"What I\u0027m suggesting is the following algebraic trick,"},{"Start":"01:29.930 ","End":"01:35.645","Text":"is to write down e to the x cubed times what I really want,"},{"Start":"01:35.645 ","End":"01:37.670","Text":"which is 3x squared."},{"Start":"01:37.670 ","End":"01:40.565","Text":"But I can\u0027t just go ahead and change the exercise,"},{"Start":"01:40.565 ","End":"01:41.900","Text":"I have to compensate,"},{"Start":"01:41.900 ","End":"01:46.720","Text":"so what I do is I also divide by 3 and this constant can be put in front."},{"Start":"01:46.720 ","End":"01:53.375","Text":"Now indeed, I do have the situation of each of the function with its derivative."},{"Start":"01:53.375 ","End":"01:56.450","Text":"The derivative of x cubed is 3x squared,"},{"Start":"01:56.450 ","End":"01:58.805","Text":"so we do fit this paradigm,"},{"Start":"01:58.805 ","End":"02:02.345","Text":"and so the answer will be 1 1/3."},{"Start":"02:02.345 ","End":"02:09.310","Text":"Now, this integral is e to the x cubed plus c, and that\u0027s it."},{"Start":"02:09.310 ","End":"02:12.245","Text":"Small trick. You don\u0027t have exactly what you want."},{"Start":"02:12.245 ","End":"02:15.590","Text":"You can often just put what you want and then make a correction."},{"Start":"02:15.590 ","End":"02:17.540","Text":"On to Part c,"},{"Start":"02:17.540 ","End":"02:21.340","Text":"integral of x over e to the 2x squared."},{"Start":"02:21.340 ","End":"02:23.480","Text":"I\u0027m sorry I misled you before."},{"Start":"02:23.480 ","End":"02:25.930","Text":"I shan\u0027t be needing this formula."},{"Start":"02:25.930 ","End":"02:28.100","Text":"Just erase it, no harm done."},{"Start":"02:28.100 ","End":"02:32.000","Text":"We only need this formula for all these 3 cases."},{"Start":"02:32.000 ","End":"02:37.955","Text":"I look at this and I don\u0027t see how I have an e to the power of something."},{"Start":"02:37.955 ","End":"02:40.160","Text":"Well, I do, but it\u0027s in the denominator."},{"Start":"02:40.160 ","End":"02:44.090","Text":"Well, that\u0027s our 1st obstacle and that\u0027s easily fixed because all you have"},{"Start":"02:44.090 ","End":"02:48.185","Text":"to do is put it in the numerator and make the exponent negative."},{"Start":"02:48.185 ","End":"02:55.625","Text":"First I have e to the minus 2x squared times x dx."},{"Start":"02:55.625 ","End":"03:00.080","Text":"Now I\u0027d like to have it as the form e to the power of f. Now,"},{"Start":"03:00.080 ","End":"03:01.550","Text":"if I look at this as f,"},{"Start":"03:01.550 ","End":"03:09.295","Text":"the derivative of the minus 2x squared is not x, it\u0027s minus 4x."},{"Start":"03:09.295 ","End":"03:13.100","Text":"Once again, we\u0027re up to playing with a constant."},{"Start":"03:13.100 ","End":"03:20.675","Text":"I\u0027m going to put the minus 4 in here and then correct by multiplying by minus a 1/4."},{"Start":"03:20.675 ","End":"03:26.840","Text":"This is equal to the integral of e to the minus 2x squared,"},{"Start":"03:26.840 ","End":"03:28.400","Text":"and I\u0027ll put what I really wanted,"},{"Start":"03:28.400 ","End":"03:30.520","Text":"which was minus 4x,"},{"Start":"03:30.520 ","End":"03:33.140","Text":"but there wasn\u0027t a minus 4 originally,"},{"Start":"03:33.140 ","End":"03:38.375","Text":"so we fix it by multiplying by minus 1 1/4."},{"Start":"03:38.375 ","End":"03:40.925","Text":"We fit the template of this."},{"Start":"03:40.925 ","End":"03:47.225","Text":"We do have e to the something and that derivative of that something alongside,"},{"Start":"03:47.225 ","End":"03:49.780","Text":"so the answer is just e to the something,"},{"Start":"03:49.780 ","End":"03:52.965","Text":"which is minus 1 1/4,"},{"Start":"03:52.965 ","End":"03:57.180","Text":"and then just this, e to the minus 2x squared,"},{"Start":"03:57.180 ","End":"04:03.430","Text":"and finally, plus c. That finishes Part c and we\u0027re done."}],"ID":8488},{"Watched":false,"Name":"Exercise 5","Duration":"4m 7s","ChapterTopicVideoID":8318,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:04.080 ","End":"00:08.340","Text":"In all 3 cases, I\u0027m going to utilize this formula"},{"Start":"00:08.340 ","End":"00:12.795","Text":"which says that if I have the integral of e to the power of a function,"},{"Start":"00:12.795 ","End":"00:15.960","Text":"and alongside it I have the derivative of that function,"},{"Start":"00:15.960 ","End":"00:18.405","Text":"then the answer is just what was written here,"},{"Start":"00:18.405 ","End":"00:20.160","Text":"e to the power of the function."},{"Start":"00:20.160 ","End":"00:29.490","Text":"Let\u0027s start with the first one which is e to the tangent x over cosine squared x."},{"Start":"00:29.490 ","End":"00:34.970","Text":"Now the first thing is that we have here a quotient and we want a product."},{"Start":"00:34.970 ","End":"00:36.754","Text":"If this is going to be our function,"},{"Start":"00:36.754 ","End":"00:38.300","Text":"possibly the tangent x,"},{"Start":"00:38.300 ","End":"00:40.040","Text":"and we want something alongside,"},{"Start":"00:40.040 ","End":"00:44.630","Text":"let\u0027s just rewrite it as e to the power of tangent x,"},{"Start":"00:44.630 ","End":"00:50.570","Text":"times 1 over cosine squared of x dx."},{"Start":"00:50.570 ","End":"00:57.140","Text":"Now, this works out fine because if we take e to the tangent x to be like e to"},{"Start":"00:57.140 ","End":"01:01.040","Text":"the f then indeed the derivative of tangent"},{"Start":"01:01.040 ","End":"01:04.970","Text":"x is 1 over cosine squared if you remember your derivatives."},{"Start":"01:04.970 ","End":"01:09.650","Text":"We fit this template so we can write the answer"},{"Start":"01:09.650 ","End":"01:16.050","Text":"as e to the power of tangent x plus c, and that\u0027s it."},{"Start":"01:16.050 ","End":"01:18.675","Text":"On to the next part."},{"Start":"01:18.675 ","End":"01:22.085","Text":"Here\u0027s part b, and in many ways it\u0027s similar to part"},{"Start":"01:22.085 ","End":"01:26.690","Text":"a in the sense that we have a quotient and we want a product so"},{"Start":"01:26.690 ","End":"01:31.515","Text":"we just write it differently as e to the arctangent of"},{"Start":"01:31.515 ","End":"01:37.170","Text":"x times 1 over 1 plus x squared."},{"Start":"01:37.170 ","End":"01:39.080","Text":"Again, if you know your formulas,"},{"Start":"01:39.080 ","End":"01:45.215","Text":"you know that the derivative of arctangent of x is exactly 1 over 1 plus x squared."},{"Start":"01:45.215 ","End":"01:47.570","Text":"Once again, we fit this template,"},{"Start":"01:47.570 ","End":"01:51.120","Text":"and the answer is just this bit here,"},{"Start":"01:51.120 ","End":"02:01.395","Text":"e to the power of arctangent of x plus c. Onto part c. Here\u0027s part c,"},{"Start":"02:01.395 ","End":"02:03.830","Text":"and the first thing I notice is that I don\u0027t have an e to"},{"Start":"02:03.830 ","End":"02:06.290","Text":"the power of something in the numerator at least."},{"Start":"02:06.290 ","End":"02:10.100","Text":"The obvious thing to do is to put this from the denominator"},{"Start":"02:10.100 ","End":"02:14.134","Text":"into the numerator by reversing the sign of the exponent."},{"Start":"02:14.134 ","End":"02:19.550","Text":"In other words, what I get is the integral of e to the power of plus"},{"Start":"02:19.550 ","End":"02:26.809","Text":"cosine 2x times sine x cosine x dx."},{"Start":"02:26.809 ","End":"02:29.960","Text":"Now I want our function f to be cosine 2x,"},{"Start":"02:29.960 ","End":"02:32.315","Text":"but first I\u0027ve got to check its derivative."},{"Start":"02:32.315 ","End":"02:37.850","Text":"If I take the derivative of cosine 2x using the chain rule,"},{"Start":"02:37.850 ","End":"02:40.445","Text":"the derivative of cosine is minus sine."},{"Start":"02:40.445 ","End":"02:43.235","Text":"I have minus sine of 2x,"},{"Start":"02:43.235 ","End":"02:47.765","Text":"but then I have to multiply by the internal derivative."},{"Start":"02:47.765 ","End":"02:50.515","Text":"The internal derivative of 2x is just 2."},{"Start":"02:50.515 ","End":"02:52.950","Text":"It\u0027s minus 2 sine 2x."},{"Start":"02:52.950 ","End":"03:00.555","Text":"Now, we have a formula that sine 2x is equal to 2 sine x cosine x,"},{"Start":"03:00.555 ","End":"03:03.330","Text":"and if I throw that into here altogether,"},{"Start":"03:03.330 ","End":"03:13.635","Text":"what I get is that the cosine 2x derivative is equal to minus 4 sine x cosine x."},{"Start":"03:13.635 ","End":"03:16.080","Text":"The difference between what I have here,"},{"Start":"03:16.080 ","End":"03:20.555","Text":"which is sine x cosine x and what I want is this minus 4."},{"Start":"03:20.555 ","End":"03:23.090","Text":"This is the usual trick that we do."},{"Start":"03:23.090 ","End":"03:25.130","Text":"We put inside what we want."},{"Start":"03:25.130 ","End":"03:30.155","Text":"We have the integral of e to the power of cosine 2x,"},{"Start":"03:30.155 ","End":"03:35.995","Text":"and then times minus 4 sine x cosine x."},{"Start":"03:35.995 ","End":"03:41.000","Text":"That is the derivative of cosine 2x at least if our computation here is correct."},{"Start":"03:41.000 ","End":"03:44.275","Text":"Then because we\u0027ve added this minus 4,"},{"Start":"03:44.275 ","End":"03:46.365","Text":"we can\u0027t just change the exercise,"},{"Start":"03:46.365 ","End":"03:48.740","Text":"we have to compensate by dividing by it,"},{"Start":"03:48.740 ","End":"03:51.610","Text":"and this we can put in front of the integral sign."},{"Start":"03:51.610 ","End":"03:57.995","Text":"Now we can use this rule with e to the cosine 2x being what\u0027s in yellow here,"},{"Start":"03:57.995 ","End":"04:01.460","Text":"and we get minus 1/4 of e to"},{"Start":"04:01.460 ","End":"04:08.710","Text":"the cosine 2x plus c. That\u0027s the answer to part c, and so we\u0027re done."}],"ID":8489},{"Watched":false,"Name":"Exercise 6","Duration":"2m 25s","ChapterTopicVideoID":8319,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.855 ","End":"00:07.275","Text":"For all 3 of them, I\u0027m going to use this formula,"},{"Start":"00:07.275 ","End":"00:11.340","Text":"which says that the integral of cosine of a function of x,"},{"Start":"00:11.340 ","End":"00:14.670","Text":"if I have the derivative of that alongside it,"},{"Start":"00:14.670 ","End":"00:16.875","Text":"cosine of f times f prime,"},{"Start":"00:16.875 ","End":"00:21.565","Text":"then the answer is just the sine of f plus the constant."},{"Start":"00:21.565 ","End":"00:26.960","Text":"The way we check it is by differentiating sine f, derivative of sine is cosine,"},{"Start":"00:26.960 ","End":"00:28.790","Text":"and this is the internal derivative."},{"Start":"00:28.790 ","End":"00:31.100","Text":"Let\u0027s take the first example"},{"Start":"00:31.100 ","End":"00:40.455","Text":"which is the integral of cosine of 2x squared plus 1 times 4x dx."},{"Start":"00:40.455 ","End":"00:44.975","Text":"Now, what I\u0027m going to take is my f would seem to be 2x squared plus 1."},{"Start":"00:44.975 ","End":"00:46.595","Text":"Let\u0027s check if this is good."},{"Start":"00:46.595 ","End":"00:48.050","Text":"We\u0027ll check f prime."},{"Start":"00:48.050 ","End":"00:52.235","Text":"F prime is the derivative of 2x squared plus 1 which is 4x."},{"Start":"00:52.235 ","End":"00:55.250","Text":"That\u0027s spot on. That\u0027s exactly what we have here."},{"Start":"00:55.250 ","End":"00:59.675","Text":"There\u0027s no problem in using this formula with f being 2x squared plus 1."},{"Start":"00:59.675 ","End":"01:06.240","Text":"The answer will then be sine of this 2x squared plus 1 plus the constant."},{"Start":"01:06.240 ","End":"01:07.955","Text":"That\u0027s all there is to it."},{"Start":"01:07.955 ","End":"01:14.055","Text":"Let\u0027s move on to part b, cosine of sine x times cosine xdx."},{"Start":"01:14.055 ","End":"01:18.995","Text":"Looking at this I would like to try that the f should be sine x."},{"Start":"01:18.995 ","End":"01:20.790","Text":"Let\u0027s just check what is f prime."},{"Start":"01:20.790 ","End":"01:23.670","Text":"If it comes out exactly this, then we\u0027re okay."},{"Start":"01:23.670 ","End":"01:27.065","Text":"Yeah, of course, it does, because the derivative of sine is cosine."},{"Start":"01:27.065 ","End":"01:29.540","Text":"So we\u0027re exactly in this situation."},{"Start":"01:29.540 ","End":"01:33.785","Text":"The answer is the sine of what was in here,"},{"Start":"01:33.785 ","End":"01:36.715","Text":"which is sine of sine x"},{"Start":"01:36.715 ","End":"01:40.455","Text":"and plus c. On to the next 1,"},{"Start":"01:40.455 ","End":"01:48.545","Text":"the integral of cosine of natural log of x all over x, dx."},{"Start":"01:48.545 ","End":"01:50.810","Text":"Well, we don\u0027t have much work to do,"},{"Start":"01:50.810 ","End":"01:52.580","Text":"but there is a little bit."},{"Start":"01:52.580 ","End":"01:58.445","Text":"You should really rewrite it so it\u0027s more visible as cosine of natural log x,"},{"Start":"01:58.445 ","End":"02:01.865","Text":"instead of divided by x times 1 over x"},{"Start":"02:01.865 ","End":"02:05.225","Text":"because then we see immediately this is f,"},{"Start":"02:05.225 ","End":"02:06.985","Text":"and this is f prime."},{"Start":"02:06.985 ","End":"02:08.430","Text":"Just a slight rewrite,"},{"Start":"02:08.430 ","End":"02:10.560","Text":"then we\u0027re in this case here."},{"Start":"02:10.560 ","End":"02:14.375","Text":"The answer is just the sine of f,"},{"Start":"02:14.375 ","End":"02:18.200","Text":"which is the sine of natural log of x,"},{"Start":"02:18.200 ","End":"02:20.395","Text":"and plus the constant."},{"Start":"02:20.395 ","End":"02:25.900","Text":"That was quick work of this useful little formula, and we\u0027re done."}],"ID":8490},{"Watched":false,"Name":"Exercise 7","Duration":"4m 32s","ChapterTopicVideoID":8320,"CourseChapterTopicPlaylistID":3989,"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":"In this exercise we have to compute the following 3 integrals,"},{"Start":"00:04.050 ","End":"00:06.360","Text":"I\u0027ve already copied the first one here,"},{"Start":"00:06.360 ","End":"00:10.050","Text":"and I\u0027ll let you know that the rule I\u0027m going to use is this one."},{"Start":"00:10.050 ","End":"00:12.645","Text":"If I have the integral of cosine of a function,"},{"Start":"00:12.645 ","End":"00:15.705","Text":"but the derivative of that function is already alongside,"},{"Start":"00:15.705 ","End":"00:18.240","Text":"then the answer is the sine of the function."},{"Start":"00:18.240 ","End":"00:20.310","Text":"Let\u0027s look if this is the case here."},{"Start":"00:20.310 ","End":"00:22.140","Text":"Here we have the cosine of something"},{"Start":"00:22.140 ","End":"00:26.235","Text":"and if this 10x to the 4th plus 1 was our f,"},{"Start":"00:26.235 ","End":"00:29.280","Text":"then what I would expect to have here was f prime."},{"Start":"00:29.280 ","End":"00:33.855","Text":"But f prime is 40x cubed,"},{"Start":"00:33.855 ","End":"00:36.705","Text":"4 from here times the 10,"},{"Start":"00:36.705 ","End":"00:38.430","Text":"so we get 40x cubed,"},{"Start":"00:38.430 ","End":"00:41.740","Text":"which is not x cubed which is what I have here."},{"Start":"00:41.740 ","End":"00:44.240","Text":"The difference is this matter of 40."},{"Start":"00:44.240 ","End":"00:46.670","Text":"This is a standard trick that we use,"},{"Start":"00:46.670 ","End":"00:48.964","Text":"we write what we want,"},{"Start":"00:48.964 ","End":"00:59.055","Text":"and what we want is the cosine of 10x to the 4th plus 1 times 40x cubed."},{"Start":"00:59.055 ","End":"01:02.640","Text":"But, we can\u0027t just go ahead and change the exercise,"},{"Start":"01:02.640 ","End":"01:04.325","Text":"this 40 wasn\u0027t there before,"},{"Start":"01:04.325 ","End":"01:06.410","Text":"so we have to compensate by dividing by 40"},{"Start":"01:06.410 ","End":"01:10.295","Text":"and we can do this in front of the integral and write 1 over 40."},{"Start":"01:10.295 ","End":"01:16.155","Text":"Now, this bit exactly fits the pattern if f is 10x to the 4th plus 1,"},{"Start":"01:16.155 ","End":"01:20.645","Text":"because now I do have this being f and this now really is f prime."},{"Start":"01:20.645 ","End":"01:24.305","Text":"I just apply the formula and what we get,"},{"Start":"01:24.305 ","End":"01:28.010","Text":"we have the 1 over 40 from here and then from the integral,"},{"Start":"01:28.010 ","End":"01:33.520","Text":"we get the sine of f which is 10x to the 4th plus 1,"},{"Start":"01:33.520 ","End":"01:35.655","Text":"and finally plus C,"},{"Start":"01:35.655 ","End":"01:38.010","Text":"and that\u0027s it for part s."},{"Start":"01:38.010 ","End":"01:39.680","Text":"Next part b."},{"Start":"01:39.680 ","End":"01:45.785","Text":"Here we have the integral of sine of x squared plus 1 times x dx,"},{"Start":"01:45.785 ","End":"01:49.850","Text":"and this rule is no longer any use to me,"},{"Start":"01:49.850 ","End":"01:53.210","Text":"I need the corresponding one for sine, for a similar,"},{"Start":"01:53.210 ","End":"02:05.070","Text":"the integral of sine of f times f prime dx is minus cosine of f plus C."},{"Start":"02:05.070 ","End":"02:07.095","Text":"Let\u0027s check."},{"Start":"02:07.095 ","End":"02:12.505","Text":"What we would like is we want our f to be x squared plus 1,"},{"Start":"02:12.505 ","End":"02:17.405","Text":"and what we need is for f prime to be alongside,"},{"Start":"02:17.405 ","End":"02:22.274","Text":"but f prime in this case is 2x,"},{"Start":"02:22.274 ","End":"02:25.700","Text":"and all I have is 1x but we know this is not a problem"},{"Start":"02:25.700 ","End":"02:32.080","Text":"because we can rewrite it as sine of x squared plus 1,"},{"Start":"02:32.080 ","End":"02:34.025","Text":"and then here put what we want,"},{"Start":"02:34.025 ","End":"02:36.915","Text":"which is 2x dx."},{"Start":"02:36.915 ","End":"02:40.350","Text":"But then, we have to compensate for multiplying by 2,"},{"Start":"02:40.350 ","End":"02:42.290","Text":"we do this by dividing by 2"},{"Start":"02:42.290 ","End":"02:44.705","Text":"which we can do in front of the integral sine."},{"Start":"02:44.705 ","End":"02:48.900","Text":"At this point we can now use what is written here,"},{"Start":"02:48.900 ","End":"02:51.810","Text":"so it\u0027s a 1/2 times minus,"},{"Start":"02:51.810 ","End":"02:54.975","Text":"let\u0027s just put the minus in front here,"},{"Start":"02:54.975 ","End":"03:00.860","Text":"so it\u0027s minus 1/2 times cosine of the f"},{"Start":"03:00.860 ","End":"03:06.140","Text":"which was x squared plus 1 and plus C."},{"Start":"03:06.140 ","End":"03:09.300","Text":"Now, onto part c."},{"Start":"03:09.300 ","End":"03:14.525","Text":"What we have here is sine of the square root of x over the square root of x."},{"Start":"03:14.525 ","End":"03:18.920","Text":"What I first of all need is a product I need sine of something times something."},{"Start":"03:18.920 ","End":"03:27.920","Text":"I\u0027ll first write it as sine square root of x times 1 over the square root of x dx."},{"Start":"03:27.920 ","End":"03:31.835","Text":"Now if I want the square root of x to be my f,"},{"Start":"03:31.835 ","End":"03:34.340","Text":"then f prime, as we know,"},{"Start":"03:34.340 ","End":"03:38.765","Text":"the derivative of square root of x is 1 over twice the square root of x."},{"Start":"03:38.765 ","End":"03:42.485","Text":"Once again we\u0027re going to have to do some adjusting with constants."},{"Start":"03:42.485 ","End":"03:44.670","Text":"What we get from here is,"},{"Start":"03:44.670 ","End":"03:51.070","Text":"this is equal to the integral of sine square root of x times 1 over"},{"Start":"03:51.070 ","End":"03:54.865","Text":"and this time I\u0027ll write twice the square root of x."},{"Start":"03:54.865 ","End":"03:57.705","Text":"Because I\u0027ve put this extra 2 in here,"},{"Start":"03:57.705 ","End":"04:01.610","Text":"and it\u0027s in the denominator I have to compensate by putting a 2 in the numerator"},{"Start":"04:01.610 ","End":"04:03.930","Text":"and it can go in front of the integral sine."},{"Start":"04:03.930 ","End":"04:07.275","Text":"Now, here I have an f and here I have f prime,"},{"Start":"04:07.275 ","End":"04:10.425","Text":"which is this bit here and this f is this bit here,"},{"Start":"04:10.425 ","End":"04:14.015","Text":"and so I have exactly this second formula here."},{"Start":"04:14.015 ","End":"04:16.490","Text":"What I have is twice,"},{"Start":"04:16.490 ","End":"04:20.165","Text":"and I need the cosine with a minus,"},{"Start":"04:20.165 ","End":"04:25.955","Text":"so it\u0027s minus twice cosine of square root of x,"},{"Start":"04:25.955 ","End":"04:27.950","Text":"and then at the end plus C."},{"Start":"04:27.950 ","End":"04:32.770","Text":"That\u0027s the end of part c, so we\u0027re done."}],"ID":8491},{"Watched":false,"Name":"Exercise 8","Duration":"2m 35s","ChapterTopicVideoID":8321,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have to compute the following integrals, there\u0027s 3 of them."},{"Start":"00:04.965 ","End":"00:07.620","Text":"In all 3, I\u0027m going to use the same formula,"},{"Start":"00:07.620 ","End":"00:12.915","Text":"which says that if I have the integral of a function times its derivative,"},{"Start":"00:12.915 ","End":"00:16.560","Text":"then the answer is 1/2 of that function squared."},{"Start":"00:16.560 ","End":"00:22.740","Text":"You can easily check this by differentiating this and you\u0027ll see that you get this."},{"Start":"00:22.740 ","End":"00:28.650","Text":"Let\u0027s start with the first where we have the integral of natural log of x over x."},{"Start":"00:28.650 ","End":"00:32.624","Text":"The first thing we see is we don\u0027t have a product, we have a quotient,"},{"Start":"00:32.624 ","End":"00:34.845","Text":"but we\u0027ve seen this sort of thing before,"},{"Start":"00:34.845 ","End":"00:39.015","Text":"and the way round this is just to write it in a different way."},{"Start":"00:39.015 ","End":"00:40.700","Text":"Instead of dividing by x,"},{"Start":"00:40.700 ","End":"00:43.130","Text":"we multiply by 1 over x."},{"Start":"00:43.130 ","End":"00:49.115","Text":"Now this fits the pattern nicely because if I take this natural log of x of my function,"},{"Start":"00:49.115 ","End":"00:52.280","Text":"then the derivative is exactly 1 over x."},{"Start":"00:52.280 ","End":"00:54.815","Text":"I do have f times f prime,"},{"Start":"00:54.815 ","End":"01:01.920","Text":"so all I need to do now is to write the answer as 1/2 natural log of x,"},{"Start":"01:01.920 ","End":"01:04.215","Text":"that\u0027s my f squared,"},{"Start":"01:04.215 ","End":"01:07.300","Text":"and to remember to add the C at the end."},{"Start":"01:07.300 ","End":"01:10.155","Text":"Let\u0027s try the next 1."},{"Start":"01:10.155 ","End":"01:12.350","Text":"Here, once again, like in part a,"},{"Start":"01:12.350 ","End":"01:15.590","Text":"we see a quotient and not a product,"},{"Start":"01:15.590 ","End":"01:17.645","Text":"so we use the same trick again."},{"Start":"01:17.645 ","End":"01:20.420","Text":"That is we write it slightly differently"},{"Start":"01:20.420 ","End":"01:27.805","Text":"as the arctangent of x times 1 over 1 plus x squared."},{"Start":"01:27.805 ","End":"01:32.210","Text":"Now if we let the arctangent of x be f of x,"},{"Start":"01:32.210 ","End":"01:34.685","Text":"then the derivative of the arctangent,"},{"Start":"01:34.685 ","End":"01:38.060","Text":"and you can look it up in your formula sheets or maybe you remember,"},{"Start":"01:38.060 ","End":"01:40.395","Text":"is exactly what\u0027s written here."},{"Start":"01:40.395 ","End":"01:42.920","Text":"We have a case of f times f prime,"},{"Start":"01:42.920 ","End":"01:47.600","Text":"in which case we can just apply the formula and say that the answer is 1/2 of"},{"Start":"01:47.600 ","End":"01:54.134","Text":"my f arctangent x squared plus constant."},{"Start":"01:54.134 ","End":"02:00.305","Text":"Now, the last 1, c. Tangent x over cosine squared x."},{"Start":"02:00.305 ","End":"02:02.270","Text":"It\u0027s the third time we\u0027ve seen that we have"},{"Start":"02:02.270 ","End":"02:05.190","Text":"a quotient instead of a product but this same trick,"},{"Start":"02:05.190 ","End":"02:07.760","Text":"it worked twice, why shouldn\u0027t it work a third time?"},{"Start":"02:07.760 ","End":"02:15.470","Text":"Just rewrite it as tangent x times 1 over cosine squared x, and here,"},{"Start":"02:15.470 ","End":"02:19.280","Text":"once again we have a case where if tangent x is my function,"},{"Start":"02:19.280 ","End":"02:23.135","Text":"the derivative of tangent is 1 over cosine squared,"},{"Start":"02:23.135 ","End":"02:27.750","Text":"so I have f times f prime exactly fits the template here,"},{"Start":"02:27.750 ","End":"02:30.590","Text":"and the answer is just 1/2 f squared."},{"Start":"02:30.590 ","End":"02:35.970","Text":"In other words, 1/2 tangent x squared plus the constant."}],"ID":8492},{"Watched":false,"Name":"Exercise 9","Duration":"3m 27s","ChapterTopicVideoID":8322,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we have to compute the following 3 integrals, a,"},{"Start":"00:03.870 ","End":"00:05.910","Text":"b, and c. In all 3 of them,"},{"Start":"00:05.910 ","End":"00:07.560","Text":"I\u0027ll be using the same formula."},{"Start":"00:07.560 ","End":"00:11.295","Text":"It\u0027s abbreviated, but f is a function of x."},{"Start":"00:11.295 ","End":"00:13.875","Text":"What we have is a square root of"},{"Start":"00:13.875 ","End":"00:17.440","Text":"this function in the denominator and its derivative in the numerator."},{"Start":"00:17.440 ","End":"00:18.735","Text":"If we have this case,"},{"Start":"00:18.735 ","End":"00:22.665","Text":"then the answer is just twice the square root of the function."},{"Start":"00:22.665 ","End":"00:25.590","Text":"Let\u0027s start with the first and that will be clearer."},{"Start":"00:25.590 ","End":"00:34.400","Text":"Let\u0027s take the integral of 2x over the square root of x squared plus 1 dx."},{"Start":"00:34.400 ","End":"00:38.885","Text":"Now I\u0027d like to have the x squared plus 1 to be my function."},{"Start":"00:38.885 ","End":"00:41.405","Text":"If x squared plus 1 is my function,"},{"Start":"00:41.405 ","End":"00:42.920","Text":"then to fit this template,"},{"Start":"00:42.920 ","End":"00:45.140","Text":"I have to have f prime in the numerator,"},{"Start":"00:45.140 ","End":"00:49.770","Text":"and I do because the derivative of f is just 2x."},{"Start":"00:49.770 ","End":"00:51.630","Text":"This indeed is f prime."},{"Start":"00:51.630 ","End":"00:54.320","Text":"What we\u0027ll have to do is follow the recipe and say that"},{"Start":"00:54.320 ","End":"00:57.230","Text":"the answer is twice the square root of my function,"},{"Start":"00:57.230 ","End":"00:58.955","Text":"which is x squared plus 1,"},{"Start":"00:58.955 ","End":"01:01.270","Text":"and at the end plus the constant."},{"Start":"01:01.270 ","End":"01:03.095","Text":"That was straightforward enough."},{"Start":"01:03.095 ","End":"01:05.750","Text":"Let\u0027s go on to the next 1."},{"Start":"01:05.750 ","End":"01:10.030","Text":"In this 1 we have the square root of something in the denominator."},{"Start":"01:10.030 ","End":"01:13.715","Text":"Let\u0027s try and see if this will work as our f of x."},{"Start":"01:13.715 ","End":"01:17.400","Text":"Let\u0027s say that f is 2 sine x."},{"Start":"01:17.400 ","End":"01:19.880","Text":"Let\u0027s see if we get f prime in the numerator."},{"Start":"01:19.880 ","End":"01:27.005","Text":"Well, f prime will equal the 2 stays and the derivative of sine x is cosine x."},{"Start":"01:27.005 ","End":"01:30.680","Text":"What I would like is 2 cosine x and all I have is cosine x."},{"Start":"01:30.680 ","End":"01:32.630","Text":"Well, you\u0027ve seen this sort of thing before."},{"Start":"01:32.630 ","End":"01:35.870","Text":"All we do is write down what we want and make an adjustment."},{"Start":"01:35.870 ","End":"01:37.985","Text":"Other words, what we do is this."},{"Start":"01:37.985 ","End":"01:41.360","Text":"We write down, I\u0027ll leave the denominator as is,"},{"Start":"01:41.360 ","End":"01:43.895","Text":"the square root of 2 sine x."},{"Start":"01:43.895 ","End":"01:49.460","Text":"This 2 sine x is going to be my f. Then I put f prime in the numerator,"},{"Start":"01:49.460 ","End":"01:52.790","Text":"which is 2 cosine x. I can\u0027t just"},{"Start":"01:52.790 ","End":"01:56.450","Text":"leave it like that because I\u0027ve gone and put this 2 in and it wasn\u0027t there before,"},{"Start":"01:56.450 ","End":"02:00.260","Text":"so we need to compensate and as usual we have to multiply by 1/2."},{"Start":"02:00.260 ","End":"02:03.385","Text":"That can come in front of the integral sign."},{"Start":"02:03.385 ","End":"02:07.820","Text":"Of course, now we have got exactly what we wanted."},{"Start":"02:07.820 ","End":"02:09.485","Text":"This is f prime."},{"Start":"02:09.485 ","End":"02:11.075","Text":"Check that below."},{"Start":"02:11.075 ","End":"02:15.470","Text":"We have exactly the template which means that the answer is equal"},{"Start":"02:15.470 ","End":"02:22.160","Text":"to 2 square root of 2 sine x plus a constant."},{"Start":"02:22.160 ","End":"02:24.765","Text":"Big pardon, I forgot the 1/2."},{"Start":"02:24.765 ","End":"02:27.075","Text":"Putting the 1/2 in here,"},{"Start":"02:27.075 ","End":"02:29.895","Text":"I end up the 1/2 canceling with the 2."},{"Start":"02:29.895 ","End":"02:33.000","Text":"I\u0027ll just erase the 1/2 times the 2, which is 1,"},{"Start":"02:33.000 ","End":"02:35.930","Text":"and the answer is just the square root of 2 sine x plus"},{"Start":"02:35.930 ","End":"02:42.680","Text":"c. In this 1 we have 1 over x square root of natural log of x."},{"Start":"02:42.680 ","End":"02:45.890","Text":"We don\u0027t have just a square root in the denominator,"},{"Start":"02:45.890 ","End":"02:49.630","Text":"and we don\u0027t even have anything in the numerator except that 1."},{"Start":"02:49.630 ","End":"02:51.770","Text":"We\u0027ve seen this kind of thing before."},{"Start":"02:51.770 ","End":"02:53.015","Text":"It\u0027s a standard trick."},{"Start":"02:53.015 ","End":"02:59.660","Text":"We write the 1 over x separately and put the x from the denominator into the numerator."},{"Start":"02:59.660 ","End":"03:03.665","Text":"Now it does look like this because we take our f"},{"Start":"03:03.665 ","End":"03:07.670","Text":"as natural log of x and we have the square root of f here,"},{"Start":"03:07.670 ","End":"03:10.755","Text":"and f prime is just 1 over x."},{"Start":"03:10.755 ","End":"03:14.330","Text":"It exactly fits if I take f as natural log of x,"},{"Start":"03:14.330 ","End":"03:20.480","Text":"which means that my answer will be twice the square root of my f,"},{"Start":"03:20.480 ","End":"03:24.695","Text":"which is natural log of x plus the constant."},{"Start":"03:24.695 ","End":"03:28.470","Text":"That\u0027s it. We\u0027ve done the whole set."}],"ID":8493},{"Watched":false,"Name":"Exercise 10","Duration":"2m 57s","ChapterTopicVideoID":8323,"CourseChapterTopicPlaylistID":3989,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"In this exercise, we have to compute"},{"Start":"00:02.130 ","End":"00:03.779","Text":"the following 3 integrals,"},{"Start":"00:03.779 ","End":"00:05.475","Text":"and in all 3 cases,"},{"Start":"00:05.475 ","End":"00:08.490","Text":"I\u0027ll be using the same formula rule"},{"Start":"00:08.490 ","End":"00:10.050","Text":"that if I have the integral"},{"Start":"00:10.050 ","End":"00:11.760","Text":"of the square root of a function"},{"Start":"00:11.760 ","End":"00:14.670","Text":"and alongside it is the derivative,"},{"Start":"00:14.670 ","End":"00:16.980","Text":"then the answer is 2/3,"},{"Start":"00:16.980 ","End":"00:19.515","Text":"that function to the power of 3 over 2."},{"Start":"00:19.515 ","End":"00:21.330","Text":"But instead of f^3 over 2,"},{"Start":"00:21.330 ","End":"00:22.860","Text":"if you don\u0027t like fractional powers,"},{"Start":"00:22.860 ","End":"00:23.700","Text":"you could always write it"},{"Start":"00:23.700 ","End":"00:25.470","Text":"as f times the square root of f."},{"Start":"00:25.470 ","End":"00:28.785","Text":"Starting with the first 1,"},{"Start":"00:28.785 ","End":"00:31.890","Text":"we have the integral of the square root"},{"Start":"00:31.890 ","End":"00:37.620","Text":"of x squared plus 1 times 2x dx."},{"Start":"00:37.620 ","End":"00:40.250","Text":"What looks like a candidate for my f"},{"Start":"00:40.250 ","End":"00:42.215","Text":"is what\u0027s under the square root sign,"},{"Start":"00:42.215 ","End":"00:44.630","Text":"x squared plus 1, and indeed,"},{"Start":"00:44.630 ","End":"00:47.195","Text":"if I take x squared plus 1 as my f,"},{"Start":"00:47.195 ","End":"00:49.460","Text":"f prime is just 2x,"},{"Start":"00:49.460 ","End":"00:51.725","Text":"so I perfectly fit the template,"},{"Start":"00:51.725 ","End":"00:55.080","Text":"and so the answer is just 2/3"},{"Start":"00:55.080 ","End":"00:58.080","Text":"times my f which is x squared"},{"Start":"00:58.080 ","End":"01:03.495","Text":"plus 1^3 over 2 plus a constant."},{"Start":"01:03.495 ","End":"01:05.750","Text":"On to the next 1."},{"Start":"01:05.750 ","End":"01:07.940","Text":"What I have now is the integral"},{"Start":"01:07.940 ","End":"01:10.100","Text":"of the square root of x cube"},{"Start":"01:10.100 ","End":"01:12.635","Text":"plus 4 times x squared dx."},{"Start":"01:12.635 ","End":"01:14.000","Text":"Obviously, we\u0027re going to try"},{"Start":"01:14.000 ","End":"01:16.670","Text":"and see if this thing will work as our f,"},{"Start":"01:16.670 ","End":"01:18.835","Text":"and I\u0027ll figure out f prime."},{"Start":"01:18.835 ","End":"01:20.570","Text":"Well, we compute this easily in our heads,"},{"Start":"01:20.570 ","End":"01:22.540","Text":"and we see that it\u0027s 3x squared."},{"Start":"01:22.540 ","End":"01:24.000","Text":"We don\u0027t have 3x squared."},{"Start":"01:24.000 ","End":"01:25.650","Text":"We only have 1x squared,"},{"Start":"01:25.650 ","End":"01:26.960","Text":"so what are we to do?"},{"Start":"01:26.960 ","End":"01:28.580","Text":"Well, the usual tricks,"},{"Start":"01:28.580 ","End":"01:29.510","Text":"we, first of all,"},{"Start":"01:29.510 ","End":"01:31.655","Text":"rewrite it as what we want,"},{"Start":"01:31.655 ","End":"01:34.800","Text":"which is the square root of x cube"},{"Start":"01:34.800 ","End":"01:37.650","Text":"plus 4 times 3x squared."},{"Start":"01:37.650 ","End":"01:39.420","Text":"That\u0027s what we want, dx."},{"Start":"01:39.420 ","End":"01:41.450","Text":"But, of course, we can\u0027t just go ahead"},{"Start":"01:41.450 ","End":"01:43.115","Text":"and multiply it by 3,"},{"Start":"01:43.115 ","End":"01:44.585","Text":"so we have to compensate,"},{"Start":"01:44.585 ","End":"01:45.770","Text":"we divide by 3."},{"Start":"01:45.770 ","End":"01:46.820","Text":"We can write the third here,"},{"Start":"01:46.820 ","End":"01:48.080","Text":"but the constant can come"},{"Start":"01:48.080 ","End":"01:49.790","Text":"in front of the integral sign."},{"Start":"01:49.790 ","End":"01:51.740","Text":"Now, it works out nicely,"},{"Start":"01:51.740 ","End":"01:54.140","Text":"because in the integral I have exactly"},{"Start":"01:54.140 ","End":"01:56.030","Text":"the square root of the function"},{"Start":"01:56.030 ","End":"01:58.595","Text":"times this thing which is f prime."},{"Start":"01:58.595 ","End":"01:59.994","Text":"I can use the rule,"},{"Start":"01:59.994 ","End":"02:02.705","Text":"and get that this is equal to 1/3."},{"Start":"02:02.705 ","End":"02:05.720","Text":"Now, what I need is 2/3"},{"Start":"02:05.720 ","End":"02:08.330","Text":"times x cube plus 4,"},{"Start":"02:08.330 ","End":"02:12.780","Text":"which is my f^3 over 2 plus C."},{"Start":"02:12.780 ","End":"02:15.915","Text":"Now, we have 1 last 1 to do."},{"Start":"02:15.915 ","End":"02:18.080","Text":"Here, we have a square root"},{"Start":"02:18.080 ","End":"02:20.315","Text":"of natural log of x over x."},{"Start":"02:20.315 ","End":"02:21.710","Text":"But what we want is a product,"},{"Start":"02:21.710 ","End":"02:22.760","Text":"not a quotient."},{"Start":"02:22.760 ","End":"02:25.430","Text":"It\u0027s the usual trick of just rewriting it"},{"Start":"02:25.430 ","End":"02:28.010","Text":"so that instead of x and the denominator,"},{"Start":"02:28.010 ","End":"02:29.990","Text":"we have 1 over x in the numerator."},{"Start":"02:29.990 ","End":"02:32.900","Text":"We get the square root of natural log of x"},{"Start":"02:32.900 ","End":"02:35.870","Text":"times 1 over x dx."},{"Start":"02:35.870 ","End":"02:38.780","Text":"Now, if we take natural log of x as our f,"},{"Start":"02:38.780 ","End":"02:41.900","Text":"we see that f prime is indeed 1 over x,"},{"Start":"02:41.900 ","End":"02:43.850","Text":"and so we fit the template,"},{"Start":"02:43.850 ","End":"02:45.925","Text":"and we can use the formula."},{"Start":"02:45.925 ","End":"02:49.470","Text":"What we get is 2/3 times"},{"Start":"02:49.470 ","End":"02:53.450","Text":"natural log of x^3 over 2"},{"Start":"02:53.450 ","End":"02:55.220","Text":"plus the constant."},{"Start":"02:55.220 ","End":"02:58.110","Text":"We\u0027re done with the whole set."}],"ID":8494}],"Thumbnail":null,"ID":3989}]

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