Trigonometric Integrals
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Trigonometric integrals using identities
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Trigonometric Integrals Using Substitution
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Integration using Trigonometric substitution
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[{"Name":"Trigonometric Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"2m 49s","ChapterTopicVideoID":8324,"CourseChapterTopicPlaylistID":1609,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this clip, I am going to talk briefly about trigonometrical integrals,"},{"Start":"00:05.085 ","End":"00:08.685","Text":"what they are, and what are the 2 main ways of solving them."},{"Start":"00:08.685 ","End":"00:11.415","Text":"Well, first of all, what is a trigonometric integral?"},{"Start":"00:11.415 ","End":"00:16.650","Text":"It\u0027s an integral of a function which contains only trigonometrical functions."},{"Start":"00:16.650 ","End":"00:22.065","Text":"For example, I could say that the integral of cosine x"},{"Start":"00:22.065 ","End":"00:27.965","Text":"over 1 minus sine squared x dx is a trigonometrical integral."},{"Start":"00:27.965 ","End":"00:34.945","Text":"Next 1, be the integral of sine to the 4th x dx contains only sine x."},{"Start":"00:34.945 ","End":"00:37.970","Text":"Products and powers and so forth are also allowed."},{"Start":"00:37.970 ","End":"00:42.230","Text":"Another example would be the integral of sine x times"},{"Start":"00:42.230 ","End":"00:47.360","Text":"cosine to the 4th x dx and another trigonometrical integral."},{"Start":"00:47.360 ","End":"00:51.530","Text":"An example of something that\u0027s not a trigonometrical integral might be integral"},{"Start":"00:51.530 ","End":"00:56.900","Text":"of x sine x dx because although it contains sine x,"},{"Start":"00:56.900 ","End":"00:58.340","Text":"it contains also x,"},{"Start":"00:58.340 ","End":"01:00.830","Text":"which is linear or polynomial,"},{"Start":"01:00.830 ","End":"01:02.435","Text":"but it\u0027s not trigonometric."},{"Start":"01:02.435 ","End":"01:06.170","Text":"Let me just put a line through this because this is not."},{"Start":"01:06.170 ","End":"01:10.505","Text":"Now, there are 2 main ways of solving trigonometric integrals."},{"Start":"01:10.505 ","End":"01:14.300","Text":"1 is solely by means of trigonometrical identities."},{"Start":"01:14.300 ","End":"01:19.070","Text":"The first method, which is using trigonometric identities only,"},{"Start":"01:19.070 ","End":"01:21.575","Text":"what we attempt to do is to get rid of"},{"Start":"01:21.575 ","End":"01:25.565","Text":"divisions and multiplications and powers because in integrals,"},{"Start":"01:25.565 ","End":"01:29.060","Text":"we don\u0027t like division and multiplication quotients and"},{"Start":"01:29.060 ","End":"01:32.750","Text":"products because there\u0027s no rules for products and quotients."},{"Start":"01:32.750 ","End":"01:34.460","Text":"This technique came to get rid of them."},{"Start":"01:34.460 ","End":"01:36.140","Text":"For example, here we tried to get rid of"},{"Start":"01:36.140 ","End":"01:40.160","Text":"this division and just have sums and differences."},{"Start":"01:40.160 ","End":"01:42.290","Text":"This method can actually be used to solve"},{"Start":"01:42.290 ","End":"01:45.155","Text":"quite a large number of trigonometric integrals."},{"Start":"01:45.155 ","End":"01:47.900","Text":"In fact, later it\u0027ll be a whole clip devoted to"},{"Start":"01:47.900 ","End":"01:51.025","Text":"just technique of using trigonometric identities."},{"Start":"01:51.025 ","End":"01:56.015","Text":"Now, the second method is called using trigonometrical substitutions."},{"Start":"01:56.015 ","End":"02:01.250","Text":"This technique also enables us to solve a wide variety of trigonometric integrals."},{"Start":"02:01.250 ","End":"02:05.240","Text":"It assumes you know the general method of integration by substitution."},{"Start":"02:05.240 ","End":"02:08.600","Text":"A trigonometric substitution is 1 where we substitute t"},{"Start":"02:08.600 ","End":"02:13.490","Text":"equals cosine x or sine x or something trigonometric,"},{"Start":"02:13.490 ","End":"02:15.980","Text":"and then we convert it into an integral in t."},{"Start":"02:15.980 ","End":"02:19.340","Text":"Solve that and then solve back from t back to x."},{"Start":"02:19.340 ","End":"02:22.115","Text":"A whole clip will also be devoted to"},{"Start":"02:22.115 ","End":"02:25.850","Text":"this technique so you should learn properly how to do it."},{"Start":"02:25.850 ","End":"02:30.485","Text":"That\u0027s mostly what I have to say except to summarize that"},{"Start":"02:30.485 ","End":"02:35.059","Text":"a trigonometric integral is an integral containing purely trigonometric functions."},{"Start":"02:35.059 ","End":"02:37.660","Text":"There are 2 methods of solving it."},{"Start":"02:37.660 ","End":"02:41.570","Text":"1, using only trigonometric identities and the other 1 using"},{"Start":"02:41.570 ","End":"02:46.730","Text":"trigonometric substitutions that we will be devoting an entire clip to each of these."},{"Start":"02:46.730 ","End":"02:49.620","Text":"That\u0027s about it for the introduction."}],"ID":8495}],"Thumbnail":null,"ID":1609},{"Name":"Trigonometric integrals using identities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Solution by Trigonometric Identities","Duration":"13m 20s","ChapterTopicVideoID":1608,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this clip, we\u0027re going to talk about how to solve"},{"Start":"00:03.090 ","End":"00:08.880","Text":"certain trigonometric integrals using trigonometrical identities."},{"Start":"00:08.880 ","End":"00:13.110","Text":"Now, although there are many trigonometrical identities,"},{"Start":"00:13.110 ","End":"00:16.860","Text":"there are certain favorite ones that crop up a lot."},{"Start":"00:16.860 ","End":"00:20.535","Text":"Mostly the exercises in trigonometric integrals"},{"Start":"00:20.535 ","End":"00:25.270","Text":"revolve around a small number of identities and I\u0027ll list them."},{"Start":"00:25.640 ","End":"00:31.440","Text":"First is the very popular identity that says that sine"},{"Start":"00:31.440 ","End":"00:39.090","Text":"squared Alpha plus cosine squared Alpha equals 1."},{"Start":"00:39.090 ","End":"00:45.945","Text":"Next we have the double angle formulae."},{"Start":"00:45.945 ","End":"00:49.845","Text":"We have sine of 2 Alpha."},{"Start":"00:49.845 ","End":"01:00.825","Text":"Next will be cosine of 2 Alpha and this is equal to 2 sine of Alpha, cosine of Alpha."},{"Start":"01:00.825 ","End":"01:08.185","Text":"Number 3, which is cosine 2 Alpha,"},{"Start":"01:08.185 ","End":"01:10.895","Text":"actually has several versions."},{"Start":"01:10.895 ","End":"01:16.140","Text":"The main version is sine squared."},{"Start":"01:16.210 ","End":"01:25.290","Text":"Sorry, it\u0027s cosine squared Alpha minus sine squared Alpha."},{"Start":"01:25.290 ","End":"01:35.290","Text":"Another version of cosine 2 Alpha is 2 Cosine squared Alpha minus 1."},{"Start":"01:35.290 ","End":"01:43.720","Text":"Third version is 1 minus 2 sine squared Alpha."},{"Start":"01:44.420 ","End":"01:48.085","Text":"All 3 are used quite commonly,"},{"Start":"01:48.085 ","End":"01:51.070","Text":"would say especially the last two,"},{"Start":"01:51.070 ","End":"01:55.970","Text":"because then we just have cosines or just have sines and not both."},{"Start":"01:56.520 ","End":"02:03.290","Text":"Also popular extremely is the last pair,"},{"Start":"02:03.290 ","End":"02:06.430","Text":"which is the square,"},{"Start":"02:06.430 ","End":"02:08.975","Text":"sine squared and cosine squared."},{"Start":"02:08.975 ","End":"02:14.585","Text":"We have that sine squared of Alpha is"},{"Start":"02:14.585 ","End":"02:21.160","Text":"equal to1/2"},{"Start":"02:21.160 ","End":"02:27.190","Text":"of 1 minus cosine of 2 Alpha."},{"Start":"02:27.440 ","End":"02:32.690","Text":"Similarly, the cosine squared of Alpha"},{"Start":"02:32.690 ","End":"02:39.305","Text":"is 1/2 of 1 plus cosine 2 Alpha."},{"Start":"02:39.305 ","End":"02:44.390","Text":"It\u0027s easy to deduce these from these 2."},{"Start":"02:44.390 ","End":"02:47.870","Text":"For example, let\u0027s take the last one,"},{"Start":"02:47.870 ","End":"02:49.760","Text":"the cosine squared Alpha."},{"Start":"02:49.760 ","End":"02:52.640","Text":"If I just isolate cosine Alpha here,"},{"Start":"02:52.640 ","End":"02:55.220","Text":"like I throw the one over to the other side,"},{"Start":"02:55.220 ","End":"02:58.490","Text":"it becomes 1 plus cosine 2 Alpha and then divide by the"},{"Start":"02:58.490 ","End":"03:03.450","Text":"2 I\u0027ve isolated cosine squared Alpha as this."},{"Start":"03:03.920 ","End":"03:11.310","Text":"Some people prefer to expand it as 1/2 minus 1/2 cosine 2 Alpha,"},{"Start":"03:11.310 ","End":"03:14.100","Text":"but that\u0027s a small consequence and here you might"},{"Start":"03:14.100 ","End":"03:17.205","Text":"write it as 1/2 plus 1/2 cosine 2 Alpha."},{"Start":"03:17.205 ","End":"03:24.900","Text":"No big difference. These all in all are the ones that we mainly use."},{"Start":"03:24.940 ","End":"03:30.570","Text":"I\u0027d like to give some examples of how we use them."},{"Start":"03:30.590 ","End":"03:33.075","Text":"I\u0027d like to add something."},{"Start":"03:33.075 ","End":"03:35.115","Text":"In the solved exercises,"},{"Start":"03:35.115 ","End":"03:37.820","Text":"all the formulas here will be used freely."},{"Start":"03:37.820 ","End":"03:40.730","Text":"But if a formula is used which is not in this list,"},{"Start":"03:40.730 ","End":"03:45.780","Text":"explicit mention will be made and it\u0027ll be done more explicitly."},{"Start":"03:46.250 ","End":"03:51.080","Text":"Now onto the first exercise. Here it is."},{"Start":"03:51.080 ","End":"03:56.470","Text":"We\u0027ll take the integral of sine x"},{"Start":"03:56.470 ","End":"04:03.330","Text":"plus cosine x, all squared dx."},{"Start":"04:03.330 ","End":"04:08.270","Text":"A warning, this has been said several times."},{"Start":"04:08.270 ","End":"04:11.090","Text":"There is no general formula for something squared."},{"Start":"04:11.090 ","End":"04:15.655","Text":"You can\u0027t say something like the integral of something squared."},{"Start":"04:15.655 ","End":"04:20.390","Text":"dx is simply that something cubed over 3."},{"Start":"04:20.390 ","End":"04:25.534","Text":"No way, I\u0027m going to erase this as quickly as I can."},{"Start":"04:25.534 ","End":"04:28.550","Text":"That does not work."},{"Start":"04:28.550 ","End":"04:33.680","Text":"What we have to do is use identities to try and get rid of the square,"},{"Start":"04:33.680 ","End":"04:35.885","Text":"which is really a multiplication."},{"Start":"04:35.885 ","End":"04:39.665","Text":"Luckily, there are lots of little algebraic rules."},{"Start":"04:39.665 ","End":"04:43.325","Text":"One of them is the following."},{"Start":"04:43.325 ","End":"04:46.445","Text":"That when we have a plus b all squared,"},{"Start":"04:46.445 ","End":"04:52.940","Text":"this is equal to a squared plus 2ab plus b squared,"},{"Start":"04:52.940 ","End":"04:57.920","Text":"or I prefer sometimes to put the b squared first and then the 2ab."},{"Start":"04:58.770 ","End":"05:01.000","Text":"This is going to get in the way,"},{"Start":"05:01.000 ","End":"05:03.320","Text":"I\u0027ll just move it,"},{"Start":"05:05.720 ","End":"05:09.730","Text":"out of the way to the side here."},{"Start":"05:11.720 ","End":"05:16.035","Text":"Now we can go ahead and expand this."},{"Start":"05:16.035 ","End":"05:24.450","Text":"This is equal to the integral of sine squared x plus cosine squared x,"},{"Start":"05:24.450 ","End":"05:26.310","Text":"which is b squared,"},{"Start":"05:26.310 ","End":"05:29.230","Text":"taking a as sine x and b is cosine x,"},{"Start":"05:29.230 ","End":"05:32.320","Text":"of course, plus twice sine x,"},{"Start":"05:32.320 ","End":"05:40.105","Text":"cosine x, and all this dx."},{"Start":"05:40.105 ","End":"05:44.075","Text":"Now look very lucky here,"},{"Start":"05:44.075 ","End":"05:49.055","Text":"I\u0027ve got two formulas that will help me a lot to simplify sine squared x plus cosine"},{"Start":"05:49.055 ","End":"05:55.320","Text":"squared x using the first formula is simply 1.. From these two,"},{"Start":"05:55.320 ","End":"05:57.570","Text":"I get just 1."},{"Start":"05:57.570 ","End":"06:00.920","Text":"From this, if I look over here,"},{"Start":"06:00.920 ","End":"06:03.050","Text":"I\u0027ll get sine of 2x."},{"Start":"06:03.050 ","End":"06:10.660","Text":"I get 1 plus sine 2x."},{"Start":"06:10.670 ","End":"06:14.320","Text":"Integral of this, of course."},{"Start":"06:15.170 ","End":"06:20.540","Text":"Now something else that will come in useful."},{"Start":"06:23.450 ","End":"06:28.600","Text":"What I meant to say was that there are a couple of formulas, not trigonometric,"},{"Start":"06:28.600 ","End":"06:33.805","Text":"they\u0027re integral formulas that are used so often."},{"Start":"06:33.805 ","End":"06:36.850","Text":"There\u0027s a couple of them, one for sine and one for cosine."},{"Start":"06:36.850 ","End":"06:39.790","Text":"One is that if instead of sine x,"},{"Start":"06:39.790 ","End":"06:44.574","Text":"I have sine of ax in an integral,"},{"Start":"06:44.574 ","End":"06:47.665","Text":"then what it is, is 1/a."},{"Start":"06:47.665 ","End":"06:52.635","Text":"It\u0027s important to remember this 1/a times minus,"},{"Start":"06:52.635 ","End":"06:57.610","Text":"we\u0027ll put the minus here, times cosine ax."},{"Start":"06:57.610 ","End":"07:00.100","Text":"You always put the C at the end."},{"Start":"07:00.100 ","End":"07:04.500","Text":"There\u0027s a similar one for cosine of ax."},{"Start":"07:05.990 ","End":"07:12.965","Text":"That is without the minus just 1 over a sine ax,"},{"Start":"07:12.965 ","End":"07:16.070","Text":"and also with a plus C at the end,"},{"Start":"07:16.070 ","End":"07:18.685","Text":"the very end of the line."},{"Start":"07:18.685 ","End":"07:21.620","Text":"Here, we can now do it immediately."},{"Start":"07:21.620 ","End":"07:24.470","Text":"We now break it up into two separate bits."},{"Start":"07:24.470 ","End":"07:28.710","Text":"For the sine 2x will be using this formula over here."},{"Start":"07:29.740 ","End":"07:33.740","Text":"Let me just get the right color."},{"Start":"07:33.740 ","End":"07:38.499","Text":"What we have here now is the integral of 1,"},{"Start":"07:38.499 ","End":"07:40.200","Text":"because there\u0027s a plus here."},{"Start":"07:40.200 ","End":"07:43.355","Text":"We separate it into the integral of this plus the integral of this."},{"Start":"07:43.355 ","End":"07:46.119","Text":"Integral of 1 is just x,"},{"Start":"07:46.119 ","End":"07:51.330","Text":"integral of sine 2x according to this is minus."},{"Start":"07:51.330 ","End":"07:56.450","Text":"Let me just change that plus into a"},{"Start":"07:56.450 ","End":"08:03.270","Text":"minus 1/2 cosine 2x."},{"Start":"08:03.270 ","End":"08:06.760","Text":"At the end we put the constant."},{"Start":"08:06.800 ","End":"08:11.945","Text":"Here\u0027s one solved example using some of these formulas,"},{"Start":"08:11.945 ","End":"08:22.765","Text":"this one and this one and also using this integration formula."},{"Start":"08:22.765 ","End":"08:26.705","Text":"Let\u0027s solve another example."},{"Start":"08:26.705 ","End":"08:30.670","Text":"I\u0027ll clear this and put the next one."},{"Start":"08:30.670 ","End":"08:33.410","Text":"Here\u0027s an example."},{"Start":"08:33.410 ","End":"08:39.220","Text":"The integral of cosine to the fourth x minus sine to the fourth x dx."},{"Start":"08:39.220 ","End":"08:44.060","Text":"As before, we\u0027ll begin with a bit of algebra that will simplify our lives."},{"Start":"08:44.060 ","End":"08:50.600","Text":"Now in algebra, there is a formula that a squared minus b"},{"Start":"08:50.600 ","End":"08:57.980","Text":"squared is equal to a plus b times a minus b."},{"Start":"08:57.980 ","End":"09:00.760","Text":"But here it doesn\u0027t look like it\u0027s to the power of 2,"},{"Start":"09:00.760 ","End":"09:02.480","Text":"it looks like the power of 4."},{"Start":"09:02.480 ","End":"09:06.290","Text":"But what if we let in that example that a is cosine"},{"Start":"09:06.290 ","End":"09:12.500","Text":"squared x and the p is sine squared of x."},{"Start":"09:12.500 ","End":"09:16.655","Text":"Then we will get a squared minus b squared."},{"Start":"09:16.655 ","End":"09:22.040","Text":"We can write this as the integral"},{"Start":"09:22.040 ","End":"09:29.200","Text":"of cosine squared plus sine squared."},{"Start":"09:29.960 ","End":"09:41.430","Text":"Then the difference cosine squared minus sine squared dx."},{"Start":"09:41.430 ","End":"09:47.290","Text":"Now look, this one is exactly this formula number 1."},{"Start":"09:47.290 ","End":"09:52.865","Text":"This thing will equal 1 and the cosine squared minus sine squared here it is,"},{"Start":"09:52.865 ","End":"09:56.190","Text":"That\u0027s cosine of 2 Alpha."},{"Start":"09:56.290 ","End":"10:05.280","Text":"What we\u0027ll get here is simply the integral of this is 1.I don\u0027t even need it."},{"Start":"10:05.280 ","End":"10:08.440","Text":"This is 1, this whole thing."},{"Start":"10:09.730 ","End":"10:14.180","Text":"Excuse me. See, I don\u0027t need to write it even this is just 1."},{"Start":"10:14.180 ","End":"10:17.780","Text":"This is, like I said, cosine of 2x."},{"Start":"10:17.780 ","End":"10:27.675","Text":"All I have to do is say that this is the integral of cosine 2x dx."},{"Start":"10:27.675 ","End":"10:32.240","Text":"Then using this formula with a is equal to 2,"},{"Start":"10:32.240 ","End":"10:35.795","Text":"this is equal to 1/2 sine 2x."},{"Start":"10:35.795 ","End":"10:43.640","Text":"Equal to 1/2 sine 2x plus the constant."},{"Start":"10:43.640 ","End":"10:46.480","Text":"That\u0027s another example."},{"Start":"10:46.480 ","End":"10:50.770","Text":"Now I think we\u0027ll go onto a third."},{"Start":"10:51.160 ","End":"10:54.810","Text":"Here\u0027s our third example."},{"Start":"10:54.980 ","End":"10:58.820","Text":"Some examples need a bit more resourcefulness."},{"Start":"10:58.820 ","End":"11:02.360","Text":"For example, here, if it\u0027s not clear immediately what to do just"},{"Start":"11:02.360 ","End":"11:06.980","Text":"start leafing through the formulas and see if anything looks like it could help."},{"Start":"11:06.980 ","End":"11:12.705","Text":"I was looking through I got to this one and I saw sine Alpha, cosine Alpha."},{"Start":"11:12.705 ","End":"11:15.170","Text":"Sure, there was a 2 here, which wasn\u0027t here,"},{"Start":"11:15.170 ","End":"11:18.455","Text":"but that\u0027s easily removed to the other side."},{"Start":"11:18.455 ","End":"11:22.455","Text":"Basically, what I figured to do was try this one,"},{"Start":"11:22.455 ","End":"11:25.020","Text":"only if I started putting the 2 here,"},{"Start":"11:25.020 ","End":"11:34.830","Text":"what I can do is let me just erase it from here and put it here as 1/2."},{"Start":"11:35.240 ","End":"11:39.875","Text":"I can take the 2 from here and put it in front of here."},{"Start":"11:39.875 ","End":"11:47.375","Text":"Now I can rewrite this in the form of 1/2,"},{"Start":"11:47.375 ","End":"11:54.380","Text":"which I can keep in front of the integral sign times now if Alpha is 3x,"},{"Start":"11:54.380 ","End":"11:56.765","Text":"then 2 Alpha is 6x."},{"Start":"11:56.765 ","End":"12:01.860","Text":"I have integral of sine of 6x dx."},{"Start":"12:04.250 ","End":"12:10.590","Text":"I hope you are following that If Alpha here is 3x,"},{"Start":"12:10.590 ","End":"12:15.645","Text":"what we have here and that the 2 Alpha is twice 3x, which is 6x."},{"Start":"12:15.645 ","End":"12:19.285","Text":"Now this, we already know how to do this integral."},{"Start":"12:19.285 ","End":"12:27.240","Text":"Cosine of 6x is like sine of ax where a is equal to 6."},{"Start":"12:27.240 ","End":"12:35.390","Text":"What we\u0027ll get is 1/2 and according to this rule,"},{"Start":"12:35.390 ","End":"12:46.680","Text":"minus 1 over 6 cosine of 6x plus"},{"Start":"12:46.680 ","End":"12:51.080","Text":"c. You should really combine these and write it as minus 1 over"},{"Start":"12:51.080 ","End":"12:59.629","Text":"12 cosine 6x plus c and that\u0027s the third example."},{"Start":"12:59.629 ","End":"13:02.390","Text":"That\u0027s the last example for this clip,"},{"Start":"13:02.390 ","End":"13:06.755","Text":"there are many more examples in the solved exercises clips,"},{"Start":"13:06.755 ","End":"13:10.010","Text":"some with different identities,"},{"Start":"13:10.010 ","End":"13:13.340","Text":"some more sophisticated, some less sophisticated, and so on."},{"Start":"13:13.340 ","End":"13:16.790","Text":"Go to it and I hope you even enjoy doing them."},{"Start":"13:16.790 ","End":"13:19.590","Text":"I\u0027m done here for now."}],"ID":1620},{"Watched":false,"Name":"Exercise 1","Duration":"3m 52s","ChapterTopicVideoID":6653,"CourseChapterTopicPlaylistID":1608,"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 integrals."},{"Start":"00:03.870 ","End":"00:06.000","Text":"I\u0027ve written a couple of formulas here."},{"Start":"00:06.000 ","End":"00:09.060","Text":"These are the ones that I\u0027ll need during the exercise."},{"Start":"00:09.060 ","End":"00:12.930","Text":"First one, sine x plus cosine x."},{"Start":"00:12.930 ","End":"00:14.340","Text":"We don\u0027t need any formula."},{"Start":"00:14.340 ","End":"00:17.220","Text":"You just have to remember the principle that when I have a sum,"},{"Start":"00:17.220 ","End":"00:20.175","Text":"I integrate each piece separately and do the sum."},{"Start":"00:20.175 ","End":"00:29.100","Text":"This equals the integral of sine x plus the integral of cosine x."},{"Start":"00:29.100 ","End":"00:33.045","Text":"It\u0027s just the principle of sums and differences."},{"Start":"00:33.045 ","End":"00:36.150","Text":"The integral of sine x is immediate,"},{"Start":"00:36.150 ","End":"00:39.560","Text":"it\u0027s known to be minus cosine x."},{"Start":"00:39.560 ","End":"00:43.010","Text":"The integral of cosine x is an immediate integral,"},{"Start":"00:43.010 ","End":"00:47.655","Text":"and that\u0027s sine x, and plus constant."},{"Start":"00:47.655 ","End":"00:50.230","Text":"That\u0027s it for a."},{"Start":"00:50.900 ","End":"00:53.985","Text":"Then on to part b, which says,"},{"Start":"00:53.985 ","End":"00:58.350","Text":"we want the integral of sine 2x minus"},{"Start":"00:58.350 ","End":"01:05.205","Text":"4 cosine of x over 3 dx."},{"Start":"01:05.205 ","End":"01:09.380","Text":"One of the principles is difference"},{"Start":"01:09.380 ","End":"01:13.250","Text":"as in sums and differences of integrals and also multiplication by a"},{"Start":"01:13.250 ","End":"01:19.610","Text":"constant which basically means that I can take the integral"},{"Start":"01:19.610 ","End":"01:28.950","Text":"separately of sine 2x minus 4 comes out of the integral sign."},{"Start":"01:29.240 ","End":"01:32.340","Text":"The difference becomes a difference,"},{"Start":"01:32.340 ","End":"01:35.280","Text":"and the 4 comes out of the integral sign."},{"Start":"01:35.280 ","End":"01:43.370","Text":"This is what we get, sine 2x the integral minus 4 times the integral of cosine x over 3."},{"Start":"01:43.370 ","End":"01:48.750","Text":"Now, I\u0027m going to use these formulas here which generalizes,"},{"Start":"01:48.750 ","End":"01:50.970","Text":"instead of x, you have ax plus b."},{"Start":"01:50.970 ","End":"01:54.890","Text":"The important thing is the coefficient of x in this,"},{"Start":"01:54.890 ","End":"01:57.125","Text":"which we call a here."},{"Start":"01:57.125 ","End":"02:03.955","Text":"In this case, I\u0027m going to use this rule with a equals 2."},{"Start":"02:03.955 ","End":"02:05.930","Text":"In the second one,"},{"Start":"02:05.930 ","End":"02:09.995","Text":"I\u0027m going to use the rule with a equals,"},{"Start":"02:09.995 ","End":"02:12.110","Text":"well, not 3 but 1 over 3,"},{"Start":"02:12.110 ","End":"02:15.300","Text":"so include the denominator here."},{"Start":"02:16.160 ","End":"02:20.990","Text":"What I will get is from here,"},{"Start":"02:20.990 ","End":"02:22.700","Text":"using the one for sine,"},{"Start":"02:22.700 ","End":"02:27.050","Text":"I have minus 1 over a,"},{"Start":"02:27.050 ","End":"02:34.640","Text":"a is 2 minus 1 over 2 times the cosine of whatever was in the brackets,"},{"Start":"02:34.640 ","End":"02:36.980","Text":"which was just 2x here."},{"Start":"02:36.980 ","End":"02:38.870","Text":"Then for the cosine,"},{"Start":"02:38.870 ","End":"02:40.655","Text":"I need this rule,"},{"Start":"02:40.655 ","End":"02:43.510","Text":"where the a is 1/3."},{"Start":"02:43.510 ","End":"02:47.960","Text":"It\u0027s this minus because it was already there"},{"Start":"02:47.960 ","End":"02:51.050","Text":"and 1 over"},{"Start":"02:51.050 ","End":"02:59.210","Text":"1/3 is 3."},{"Start":"02:59.210 ","End":"03:01.560","Text":"That\u0027s the 3, which is 1/3."},{"Start":"03:01.560 ","End":"03:04.070","Text":"Sine of whatever was here,"},{"Start":"03:04.070 ","End":"03:07.855","Text":"which is x over 3 plus constant."},{"Start":"03:07.855 ","End":"03:11.545","Text":"This is our answer to part b."},{"Start":"03:11.545 ","End":"03:19.920","Text":"Part c is just the integral of sine 0.5xdx."},{"Start":"03:19.920 ","End":"03:31.035","Text":"Once again, I\u0027m going to use this formula with my a equaling, it\u0027s highlighted, 0.5."},{"Start":"03:31.035 ","End":"03:35.805","Text":"What I get is the minus from here."},{"Start":"03:35.805 ","End":"03:40.845","Text":"1 over a, 1 over 0.5 is 2,"},{"Start":"03:40.845 ","End":"03:43.740","Text":"and cosine of whatever this was,"},{"Start":"03:43.740 ","End":"03:53.650","Text":"which is 0.5x and plus c. Let\u0027s finish last part, and this set."}],"ID":6712},{"Watched":false,"Name":"Exercise 2","Duration":"2m 47s","ChapterTopicVideoID":6654,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.975 ","End":"00:07.890","Text":"In the first 1, we just have to use the difference rule that the"},{"Start":"00:07.890 ","End":"00:12.480","Text":"integral of this is the difference of the integrals."},{"Start":"00:12.480 ","End":"00:20.010","Text":"What I have is that the integral of 1 over sine squared x minus"},{"Start":"00:20.010 ","End":"00:28.725","Text":"1 over cosine squared x dx is equal to 2 separate bits."},{"Start":"00:28.725 ","End":"00:36.500","Text":"First of all, the integral of 1 over sine squared x dx,"},{"Start":"00:36.500 ","End":"00:41.135","Text":"and then subtract the integral of 1 over cosine"},{"Start":"00:41.135 ","End":"00:47.445","Text":"squared x dx, so subtract."},{"Start":"00:47.445 ","End":"00:53.270","Text":"Now, each of these is an immediate integral because the integral of 1"},{"Start":"00:53.270 ","End":"01:02.970","Text":"over sine squared is minus cotangent x,"},{"Start":"01:02.970 ","End":"01:06.870","Text":"and the integral of 1 over cosine squared is just tangent."},{"Start":"01:06.870 ","End":"01:08.415","Text":"But there\u0027s a minus here,"},{"Start":"01:08.415 ","End":"01:15.475","Text":"so it\u0027s minus tangent x plus c. That\u0027s the answer to Part A."},{"Start":"01:15.475 ","End":"01:19.560","Text":"Onto Part B, these 2 rules,"},{"Start":"01:19.560 ","End":"01:24.590","Text":"we use the immediate ones where this was minus cotangent and this was tangent,"},{"Start":"01:24.590 ","End":"01:26.210","Text":"it can actually be generalized."},{"Start":"01:26.210 ","End":"01:28.820","Text":"Instead of x, if we had ax plus b,"},{"Start":"01:28.820 ","End":"01:31.820","Text":"then we just have a 1 over a in front of"},{"Start":"01:31.820 ","End":"01:36.305","Text":"the tangent and in front of the minus cotangent respectively."},{"Start":"01:36.305 ","End":"01:41.370","Text":"In b, we happen to have the 1 over cosine squared,"},{"Start":"01:41.370 ","End":"01:44.200","Text":"and a is 4 here."},{"Start":"01:44.240 ","End":"01:54.260","Text":"All I have to do is say that the integral of 1 over cosine squared 4x dx,"},{"Start":"01:54.260 ","End":"02:03.510","Text":"where my a is equal to 4 is just equal to the 1 over 4,"},{"Start":"02:03.510 ","End":"02:08.240","Text":"and then times the tangent or whatever is in the brackets,"},{"Start":"02:08.240 ","End":"02:10.150","Text":"in this case just 4x,"},{"Start":"02:10.150 ","End":"02:15.960","Text":"and plus c. Similarly, for the last 1,"},{"Start":"02:15.960 ","End":"02:25.065","Text":"we have the integral of 1 over sine squared 10x."},{"Start":"02:25.065 ","End":"02:29.159","Text":"Here our a is equal to 10,"},{"Start":"02:29.159 ","End":"02:33.659","Text":"so we start out with 1 over 10,"},{"Start":"02:33.659 ","End":"02:35.690","Text":"there\u0027s a minus in the formula,"},{"Start":"02:35.690 ","End":"02:38.195","Text":"so it\u0027s minus 1 over 10,"},{"Start":"02:38.195 ","End":"02:42.810","Text":"and cotangent of whatever it was,"},{"Start":"02:42.810 ","End":"02:48.160","Text":"x plus b was 10x plus c. That\u0027s it."}],"ID":6713},{"Watched":false,"Name":"Exercise 3","Duration":"5m 56s","ChapterTopicVideoID":6655,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have to compute the following integrals, a,"},{"Start":"00:03.630 ","End":"00:05.940","Text":"b, and c. Let\u0027s start with a,"},{"Start":"00:05.940 ","End":"00:08.040","Text":"I\u0027ve copied it already."},{"Start":"00:08.040 ","End":"00:14.220","Text":"I brought a formula with me which talks about cosine squared minus sine squared,"},{"Start":"00:14.220 ","End":"00:15.600","Text":"but I have the opposite."},{"Start":"00:15.600 ","End":"00:17.820","Text":"I have sine squared minus cosine squared."},{"Start":"00:17.820 ","End":"00:21.345","Text":"Well, that\u0027s no problem because from basic algebra,"},{"Start":"00:21.345 ","End":"00:24.000","Text":"if I put a minus in front of a difference,"},{"Start":"00:24.000 ","End":"00:26.190","Text":"I can reverse the order of the difference."},{"Start":"00:26.190 ","End":"00:29.880","Text":"This is exactly equal to minus the"},{"Start":"00:29.880 ","End":"00:35.745","Text":"integral of the other way around of the cosine squared minus sine squared."},{"Start":"00:35.745 ","End":"00:37.910","Text":"Now, what do I do with this?"},{"Start":"00:37.910 ","End":"00:40.470","Text":"Well, I\u0027ve presented a formula here."},{"Start":"00:40.470 ","End":"00:50.985","Text":"This is equal 2 just by putting cosine 2x minus the integral of cosine 2x dx."},{"Start":"00:50.985 ","End":"00:54.860","Text":"Now, I know what the integral of cosine x is,"},{"Start":"00:54.860 ","End":"01:03.335","Text":"but cosine 2x is one of these generalizations where we say that the integral of cosine,"},{"Start":"01:03.335 ","End":"01:09.140","Text":"you could take ax plus b or we\u0027ll just take ax dx."},{"Start":"01:09.140 ","End":"01:10.550","Text":"If it was just cosine,"},{"Start":"01:10.550 ","End":"01:12.530","Text":"the answer would be sine,"},{"Start":"01:12.530 ","End":"01:16.325","Text":"but it\u0027s generalized and we have a 1 over a in front."},{"Start":"01:16.325 ","End":"01:19.970","Text":"In other words, when we have x, ax,"},{"Start":"01:19.970 ","End":"01:24.650","Text":"or ax plus b, then we have to divide by this a."},{"Start":"01:24.650 ","End":"01:28.100","Text":"In which case what we get here is"},{"Start":"01:28.100 ","End":"01:32.990","Text":"minus from the minus and now the integral from the cosine,"},{"Start":"01:32.990 ","End":"01:35.915","Text":"we get 1/a, which is 1/2."},{"Start":"01:35.915 ","End":"01:41.060","Text":"We have now minus a half sine of the same ax,"},{"Start":"01:41.060 ","End":"01:44.850","Text":"which is 2x plus c constant."},{"Start":"01:44.850 ","End":"01:48.960","Text":"Okay. That is the answer to part a."},{"Start":"01:48.960 ","End":"01:51.820","Text":"Now, part b."},{"Start":"01:52.640 ","End":"01:58.400","Text":"Here\u0027s exercise b. I don\u0027t think I need all of the formulas here."},{"Start":"01:58.400 ","End":"01:59.420","Text":"I\u0027ll erase some."},{"Start":"01:59.420 ","End":"02:01.794","Text":"This one I don\u0027t need,"},{"Start":"02:01.794 ","End":"02:04.755","Text":"but instead, I will need another one."},{"Start":"02:04.755 ","End":"02:09.225","Text":"I\u0027ll need the formula for a^4 minus b^4,"},{"Start":"02:09.225 ","End":"02:14.779","Text":"which was really just a generalization of the a squared minus b squared formula."},{"Start":"02:14.779 ","End":"02:20.479","Text":"It\u0027s a squared minus b squared,"},{"Start":"02:20.479 ","End":"02:23.390","Text":"a squared plus b squared."},{"Start":"02:23.390 ","End":"02:28.505","Text":"It\u0027s just a generalization of the a squared minus b squared formula,"},{"Start":"02:28.505 ","End":"02:31.940","Text":"but with little a squared instead of a and so on."},{"Start":"02:31.940 ","End":"02:33.815","Text":"Finally, in case you forgotten,"},{"Start":"02:33.815 ","End":"02:36.940","Text":"I\u0027ll need the old classic trigonometric formula,"},{"Start":"02:36.940 ","End":"02:41.530","Text":"sine squared x plus cosine squared x equals one."},{"Start":"02:41.530 ","End":"02:45.045","Text":"I think this should do if I need more, we\u0027ll see."},{"Start":"02:45.045 ","End":"02:46.655","Text":"I, first of all, write this,"},{"Start":"02:46.655 ","End":"02:56.560","Text":"I\u0027ll expand using the algebra and I\u0027ll get the integral of cosine squared x minus sine"},{"Start":"02:56.560 ","End":"03:01.940","Text":"squared x times cosine"},{"Start":"03:01.940 ","End":"03:09.195","Text":"squared x plus sine squared x; all this, dx."},{"Start":"03:09.195 ","End":"03:11.920","Text":"Now look, this plus formula here means that"},{"Start":"03:11.920 ","End":"03:16.925","Text":"this whole thing is equal to 1 and therefore can be dismissed."},{"Start":"03:16.925 ","End":"03:20.300","Text":"Once more cosine squared minus sine squared,"},{"Start":"03:20.300 ","End":"03:24.170","Text":"just like in the previous exercise is cosine 2x."},{"Start":"03:24.170 ","End":"03:30.795","Text":"All I have left is the integral of cosine 2x dx."},{"Start":"03:30.795 ","End":"03:32.300","Text":"Although we did it in part a,"},{"Start":"03:32.300 ","End":"03:37.925","Text":"I\u0027ll just do it again because we still have this formula here with a being 2,"},{"Start":"03:37.925 ","End":"03:41.795","Text":"so we put it as 1/2 from here,"},{"Start":"03:41.795 ","End":"03:44.540","Text":"sine of the same ax,"},{"Start":"03:44.540 ","End":"03:48.030","Text":"sine 2x plus c. Now,"},{"Start":"03:48.030 ","End":"03:49.740","Text":"we come to part c,"},{"Start":"03:49.740 ","End":"03:53.190","Text":"which is the integral of"},{"Start":"03:53.190 ","End":"04:00.735","Text":"sine x plus cosine x, all squared."},{"Start":"04:00.735 ","End":"04:05.484","Text":"I\u0027m going to need some different formulas from these I can see already,"},{"Start":"04:05.484 ","End":"04:07.970","Text":"but this will still come in useful."},{"Start":"04:07.970 ","End":"04:10.675","Text":"Let me write down the formulas that we need."},{"Start":"04:10.675 ","End":"04:17.480","Text":"I got all the formulas I needed and I\u0027m going to start with the first one which I could"},{"Start":"04:17.480 ","End":"04:24.700","Text":"just change the order and write it as a squared plus b squared plus 2ab."},{"Start":"04:24.700 ","End":"04:26.690","Text":"The order doesn\u0027t matter,"},{"Start":"04:26.690 ","End":"04:31.985","Text":"it\u0027s just more convenient because then in the beginning I get cosine"},{"Start":"04:31.985 ","End":"04:39.050","Text":"squared x plus sine squared x and that\u0027s very good for me because that\u0027s equal to 1,"},{"Start":"04:39.050 ","End":"04:40.415","Text":"then we\u0027ll get it written it here."},{"Start":"04:40.415 ","End":"04:44.990","Text":"Then plus the 2ab plus 2 sine x,"},{"Start":"04:44.990 ","End":"04:50.650","Text":"cosine x, and put that in brackets, dx."},{"Start":"04:50.650 ","End":"04:56.510","Text":"We see, the first 2 terms is the sine squared plus cosine squared,"},{"Start":"04:56.510 ","End":"04:59.860","Text":"so that\u0027s 1 and the last term,"},{"Start":"04:59.860 ","End":"05:04.775","Text":"which is the product is exactly what\u0027s written here,"},{"Start":"05:04.775 ","End":"05:06.905","Text":"so that\u0027s sine 2x."},{"Start":"05:06.905 ","End":"05:11.260","Text":"What I have is the integral of 1 plus sine 2x."},{"Start":"05:11.260 ","End":"05:13.070","Text":"That\u0027s the sum of 2 integrals."},{"Start":"05:13.070 ","End":"05:20.130","Text":"Integral of 1 is just x and the integral of sine 2x,"},{"Start":"05:20.130 ","End":"05:27.080","Text":"I can make use of this formula with 2 playing the role of a."},{"Start":"05:27.740 ","End":"05:31.550","Text":"Well, it\u0027s not really a plus, it\u0027s a minus."},{"Start":"05:31.550 ","End":"05:33.500","Text":"I can write plus negative a half."},{"Start":"05:33.500 ","End":"05:39.290","Text":"I prefer to just write it straight away as minus."},{"Start":"05:39.290 ","End":"05:47.010","Text":"I have minus the 1 over a which is 1/2 and then the cosine of the same ax,"},{"Start":"05:47.010 ","End":"05:52.335","Text":"which is 2x, and finally, plus constant."},{"Start":"05:52.335 ","End":"05:57.010","Text":"That\u0027s part c and that\u0027s the end."}],"ID":6714},{"Watched":false,"Name":"Exercise 4","Duration":"6m 50s","ChapterTopicVideoID":6656,"CourseChapterTopicPlaylistID":1608,"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":"In this exercise, we have to compute the following integrals as 3 of them,"},{"Start":"00:04.020 ","End":"00:06.900","Text":"a, b, and c. We\u0027ll start with a,"},{"Start":"00:06.900 ","End":"00:08.790","Text":"I\u0027ve already copied it."},{"Start":"00:08.790 ","End":"00:13.635","Text":"The formulas I\u0027ll need essentially will be these 2,"},{"Start":"00:13.635 ","End":"00:15.915","Text":"except that this one,"},{"Start":"00:15.915 ","End":"00:20.670","Text":"I prefer to write in a different form with the 2 on the other side."},{"Start":"00:20.670 ","End":"00:25.049","Text":"Instead of this, I will take the formula,"},{"Start":"00:25.049 ","End":"00:33.455","Text":"sine Alpha cosine Alpha is 1.5 sine 2 Alpha,"},{"Start":"00:33.455 ","End":"00:35.750","Text":"same as this, just divide it by 2."},{"Start":"00:35.750 ","End":"00:40.535","Text":"More convenient. Let\u0027s look here,"},{"Start":"00:40.535 ","End":"00:46.310","Text":"and we see right away that these 2 fit this formula here."},{"Start":"00:46.310 ","End":"00:50.360","Text":"I can write this as 1.5 sine 2 Alpha."},{"Start":"00:50.360 ","End":"00:52.190","Text":"The 1.5 is a constant,"},{"Start":"00:52.190 ","End":"00:56.335","Text":"so I can actually put it in front of the integral sign."},{"Start":"00:56.335 ","End":"00:57.945","Text":"Instead of this now,"},{"Start":"00:57.945 ","End":"01:02.340","Text":"I put sine, and I\u0027m writing Alpha be x of course,"},{"Start":"01:02.340 ","End":"01:11.340","Text":"so I have sine 2x and then there\u0027s also a cosine 2x that was there, dx."},{"Start":"01:11.340 ","End":"01:12.780","Text":"I meant x here,"},{"Start":"01:12.780 ","End":"01:15.040","Text":"not Alpha, excuse me."},{"Start":"01:15.040 ","End":"01:17.720","Text":"Now if we look at this,"},{"Start":"01:17.720 ","End":"01:21.120","Text":"it is sine Alpha cosine Alpha."},{"Start":"01:21.120 ","End":"01:24.560","Text":"If I let Alpha be 2x,"},{"Start":"01:25.330 ","End":"01:27.710","Text":"then I have this formula,"},{"Start":"01:27.710 ","End":"01:29.465","Text":"sine Alpha cosine Alpha."},{"Start":"01:29.465 ","End":"01:33.180","Text":"Again it\u0027s equal to 1.5 sine 2 Alpha."},{"Start":"01:33.320 ","End":"01:37.440","Text":"The 1.5 can come in front of the integral,"},{"Start":"01:37.440 ","End":"01:40.465","Text":"so I get 1.5 times 1.5."},{"Start":"01:40.465 ","End":"01:44.345","Text":"Now I have sine of 2 Alpha,"},{"Start":"01:44.345 ","End":"01:47.880","Text":"but Alpha is 2x so 2 Alpha is 4x."},{"Start":"01:48.590 ","End":"01:51.195","Text":"Alpha is 2x."},{"Start":"01:51.195 ","End":"01:53.695","Text":"Twice Alpha is 4x."},{"Start":"01:53.695 ","End":"01:58.355","Text":"Now, I can use this formula which we\u0027ve seen several times,"},{"Start":"01:58.355 ","End":"02:00.620","Text":"where instead of x,"},{"Start":"02:00.620 ","End":"02:03.875","Text":"we have ax or sometimes ax plus b,"},{"Start":"02:03.875 ","End":"02:09.285","Text":"and in which case we divide what we\u0027d expect by a."},{"Start":"02:09.285 ","End":"02:15.395","Text":"In this case, I\u0027m going to get when a is 4 minus a quarter."},{"Start":"02:15.395 ","End":"02:19.460","Text":"Let\u0027s think of this. We have a half and a half and minus a quarter."},{"Start":"02:19.460 ","End":"02:23.160","Text":"I make that minus 1 over 16 altogether."},{"Start":"02:23.160 ","End":"02:26.420","Text":"Then cosine of ax,"},{"Start":"02:26.420 ","End":"02:28.600","Text":"which is the cosine of 4x,"},{"Start":"02:28.600 ","End":"02:33.239","Text":"and that\u0027s the answer except for the constant."},{"Start":"02:33.450 ","End":"02:38.630","Text":"That does a, now on to b. I\u0027ve copied"},{"Start":"02:38.630 ","End":"02:44.795","Text":"this exercise here and I\u0027ve written a couple of formulas that will help us."},{"Start":"02:44.795 ","End":"02:47.450","Text":"Actually, I should have written Alpha here."},{"Start":"02:47.450 ","End":"02:50.730","Text":"It\u0027s a standard trigonometrical formula."},{"Start":"02:50.730 ","End":"02:54.650","Text":"We also have a formula with one of the immediate ones that the"},{"Start":"02:54.650 ","End":"02:59.150","Text":"integral of 1 over cosine squared x is tangent x."},{"Start":"02:59.150 ","End":"03:01.295","Text":"Now applying it here,"},{"Start":"03:01.295 ","End":"03:03.200","Text":"what we can do first of all,"},{"Start":"03:03.200 ","End":"03:08.165","Text":"is to write tangent squared as 1 over cosine squared minus 1,"},{"Start":"03:08.165 ","End":"03:10.100","Text":"letting Alpha equal x."},{"Start":"03:10.100 ","End":"03:18.795","Text":"What we have is the integral of 1 over cosine squared x minus 1."},{"Start":"03:18.795 ","End":"03:27.255","Text":"All this dx, and then we have 2 immediate integrals."},{"Start":"03:27.255 ","End":"03:29.150","Text":"1 over cosine squared,"},{"Start":"03:29.150 ","End":"03:31.385","Text":"it says here is tangent x."},{"Start":"03:31.385 ","End":"03:34.115","Text":"So we\u0027ll write this as tangent x."},{"Start":"03:34.115 ","End":"03:39.034","Text":"Now we have a difference minus the integral of 1 is just x,"},{"Start":"03:39.034 ","End":"03:41.705","Text":"and finally plus c,"},{"Start":"03:41.705 ","End":"03:43.970","Text":"and that does Part B."},{"Start":"03:43.970 ","End":"03:49.350","Text":"Next onto Part C. I\u0027ve copied the Part c,"},{"Start":"03:49.350 ","End":"03:52.340","Text":"and we\u0027re going to need some different formulas."},{"Start":"03:52.340 ","End":"03:57.980","Text":"One of the main ones will be the sine x times"},{"Start":"03:57.980 ","End":"04:05.265","Text":"cosine x is equal to 1/2 sine 2x."},{"Start":"04:05.265 ","End":"04:11.210","Text":"It may not look immediately familiar to you because it\u0027s usually written in another form"},{"Start":"04:11.210 ","End":"04:20.005","Text":"as 2 sine x. Cosine x is equal to sine 2x."},{"Start":"04:20.005 ","End":"04:24.065","Text":"But sometimes this form is more convenient when we don\u0027t have the 2 here,"},{"Start":"04:24.065 ","End":"04:28.535","Text":"we just throw it over to the other side as a half."},{"Start":"04:28.535 ","End":"04:38.030","Text":"That\u0027s one formula I\u0027ll be needing is the integral of 1 over sine squared."},{"Start":"04:38.030 ","End":"04:40.855","Text":"Now, sine squared x is familiar,"},{"Start":"04:40.855 ","End":"04:46.895","Text":"but there\u0027s also a generalized where it\u0027s sine squared of ax plus b or just ax."},{"Start":"04:46.895 ","End":"04:50.335","Text":"There wasn\u0027t the a here would be minus the cotangent."},{"Start":"04:50.335 ","End":"04:52.345","Text":"But since there is an a here,"},{"Start":"04:52.345 ","End":"04:55.410","Text":"we also add this 1 over a,"},{"Start":"04:55.410 ","End":"04:58.900","Text":"1 over the internal derivative provided it\u0027s constant."},{"Start":"04:58.900 ","End":"05:08.910","Text":"Then we have the cotangent of ax and plus the constant at the end. Let\u0027s get to it."},{"Start":"05:08.910 ","End":"05:12.090","Text":"Using the formula here,"},{"Start":"05:12.090 ","End":"05:17.200","Text":"what I get is that this is equal to 1 over"},{"Start":"05:17.200 ","End":"05:23.820","Text":"and here I have 1/2 of sine 2x, all squared."},{"Start":"05:23.820 ","End":"05:30.700","Text":"This equals, now if I take 1 over 1/2 squared,"},{"Start":"05:30.700 ","End":"05:35.425","Text":"basically what I get is 1 over 1/2"},{"Start":"05:35.425 ","End":"05:43.550","Text":"squared times sine squared 2x dx."},{"Start":"05:43.550 ","End":"05:45.985","Text":"1/2 squared is a quarter,"},{"Start":"05:45.985 ","End":"05:47.580","Text":"but it\u0027s on the denominator."},{"Start":"05:47.580 ","End":"05:50.545","Text":"If I bring it to the numerator, it becomes 4."},{"Start":"05:50.545 ","End":"05:54.580","Text":"The 4 I can take outside of the integral sign."},{"Start":"05:54.580 ","End":"05:59.290","Text":"I\u0027m left with 4 times the integral of 1"},{"Start":"05:59.290 ","End":"06:05.400","Text":"over sine squared 2x dx."},{"Start":"06:05.400 ","End":"06:12.205","Text":"Now I\u0027ll be wanting to use this formula here with the sine squared in the denominator."},{"Start":"06:12.205 ","End":"06:14.830","Text":"If I just let my a equal 2,"},{"Start":"06:14.830 ","End":"06:18.654","Text":"what I\u0027ll get will be I still have a 4,"},{"Start":"06:18.654 ","End":"06:22.285","Text":"but now this integral becomes minus 1 over a,"},{"Start":"06:22.285 ","End":"06:25.120","Text":"which is minus 1 over 2."},{"Start":"06:25.120 ","End":"06:28.210","Text":"Then times the cotangent of whatever was here,"},{"Start":"06:28.210 ","End":"06:32.805","Text":"which is 2x plus a constant."},{"Start":"06:32.805 ","End":"06:35.440","Text":"Finally, just to be nice, tidy it up."},{"Start":"06:35.440 ","End":"06:39.760","Text":"4 times minus a half would be minus 2."},{"Start":"06:39.760 ","End":"06:46.050","Text":"Minus 2 cotangent of 2x plus constant."},{"Start":"06:46.050 ","End":"06:51.110","Text":"That\u0027s the answer to Part C, and we\u0027re done."}],"ID":6715},{"Watched":false,"Name":"Exercise 5 Parts a-b","Duration":"6m 17s","ChapterTopicVideoID":6657,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this exercise, we have to compute the following 3 integrals."},{"Start":"00:04.335 ","End":"00:06.720","Text":"In the first 1, we have a product,"},{"Start":"00:06.720 ","End":"00:09.855","Text":"and product is no good to us."},{"Start":"00:09.855 ","End":"00:13.125","Text":"In integration, we prefer sums and differences."},{"Start":"00:13.125 ","End":"00:16.530","Text":"Luckily, there\u0027s a formula in trigonometry."},{"Start":"00:16.530 ","End":"00:20.310","Text":"It\u0027s from the formula sheets or books of trigonometry,"},{"Start":"00:20.310 ","End":"00:25.065","Text":"which tells us how to convert this product into a sum."},{"Start":"00:25.065 ","End":"00:30.870","Text":"Let\u0027s apply it where Alpha is 7x and Beta is 5x,"},{"Start":"00:30.870 ","End":"00:39.470","Text":"we get the 1/2 we can put in front of the integral sign."},{"Start":"00:39.470 ","End":"00:42.604","Text":"We have sine Alpha plus Beta,"},{"Start":"00:42.604 ","End":"00:46.685","Text":"which is sine of 12x because 7 plus 5 is 12."},{"Start":"00:46.685 ","End":"00:52.520","Text":"Sine of 12x, plus sine of Alpha"},{"Start":"00:52.520 ","End":"00:58.440","Text":"minus Beta plus sine of 2x dx."},{"Start":"00:58.440 ","End":"01:01.215","Text":"Now, this is a sum,"},{"Start":"01:01.215 ","End":"01:04.625","Text":"we can separate it into 2 separate bits."},{"Start":"01:04.625 ","End":"01:12.030","Text":"We get the 1/2 sine of 12x integral,"},{"Start":"01:12.030 ","End":"01:21.735","Text":"sine of 12x dx plus 1/2 integral of sine 2x dx."},{"Start":"01:21.735 ","End":"01:24.300","Text":"It\u0027s not sine x, sine 12x,"},{"Start":"01:24.300 ","End":"01:27.080","Text":"or sine 2x, but we also have a formula for that."},{"Start":"01:27.080 ","End":"01:30.140","Text":"Instead of x, we have ax or ax plus b,"},{"Start":"01:30.140 ","End":"01:32.210","Text":"we just divide by a,"},{"Start":"01:32.210 ","End":"01:34.480","Text":"what we would normally do."},{"Start":"01:34.480 ","End":"01:39.365","Text":"Integral of sine would normally be minus cosine."},{"Start":"01:39.365 ","End":"01:41.150","Text":"But because of the 12,"},{"Start":"01:41.150 ","End":"01:42.740","Text":"we divide by 12,"},{"Start":"01:42.740 ","End":"01:45.560","Text":"we get 1/2 times,"},{"Start":"01:45.560 ","End":"01:51.555","Text":"now, it\u0027s minus 1 over 12 cosine of 12 x."},{"Start":"01:51.555 ","End":"01:57.795","Text":"For the second 1, we also get a minus 1 over 2 minus,"},{"Start":"01:57.795 ","End":"02:01.190","Text":"we had the 1/2 originally and we have another,"},{"Start":"02:01.190 ","End":"02:10.640","Text":"let\u0027s write it as plus and then times minus 1 over 2 from this cosine 2x plus"},{"Start":"02:10.640 ","End":"02:20.210","Text":"c. That\u0027s fine except that we might want to simplify and write this as minus 1 over 24."},{"Start":"02:20.210 ","End":"02:23.000","Text":"We might want to write this as minus 1 over 4,"},{"Start":"02:23.000 ","End":"02:28.960","Text":"but other than that we can settle with this for an answer."},{"Start":"02:29.090 ","End":"02:33.180","Text":"That was part A, next part B."},{"Start":"02:33.180 ","End":"02:37.005","Text":"Part B, I\u0027ve now written here."},{"Start":"02:37.005 ","End":"02:41.330","Text":"This formula was leftover from the previous part A is"},{"Start":"02:41.330 ","End":"02:45.185","Text":"not going to be good for us because this deals with sine times cosine."},{"Start":"02:45.185 ","End":"02:49.925","Text":"We need the other 2, the cosine times cosine and the sine times sine."},{"Start":"02:49.925 ","End":"02:51.835","Text":"I\u0027ll just write them in here."},{"Start":"02:51.835 ","End":"02:54.770","Text":"These are the 2 formulas that we need."},{"Start":"02:54.770 ","End":"02:57.465","Text":"The first one, we\u0027ll use for the product of cosines,"},{"Start":"02:57.465 ","End":"03:00.620","Text":"and the second one, for productive signs."},{"Start":"03:00.620 ","End":"03:08.300","Text":"What we\u0027ll get is the integral k. Let\u0027s first of all deal with the first bit,"},{"Start":"03:08.300 ","End":"03:12.620","Text":"the cosine times cosine using this rule here,"},{"Start":"03:12.620 ","End":"03:17.235","Text":"where Alpha will be x and Beta will be 2x."},{"Start":"03:17.235 ","End":"03:26.470","Text":"What we\u0027ll get is 1/2 cosine of Alpha plus Beta is cosine 3x"},{"Start":"03:26.470 ","End":"03:37.635","Text":"plus cosine of Alpha minus Beta will be cosine of minus x."},{"Start":"03:37.635 ","End":"03:40.830","Text":"That\u0027s the first part."},{"Start":"03:40.830 ","End":"03:44.490","Text":"Then the second part, this part,"},{"Start":"03:44.490 ","End":"03:47.345","Text":"we\u0027ll take using this formula again,"},{"Start":"03:47.345 ","End":"03:51.180","Text":"Alpha is x and Beta is 2x."},{"Start":"03:51.260 ","End":"03:55.745","Text":"What we get is from here,"},{"Start":"03:55.745 ","End":"04:01.805","Text":"the integral of the sum, plus 1/2 again."},{"Start":"04:01.805 ","End":"04:05.690","Text":"From here we get cosine of Alpha minus Beta,"},{"Start":"04:05.690 ","End":"04:09.090","Text":"which is cosine of minus x,"},{"Start":"04:11.090 ","End":"04:16.355","Text":"and then minus from here,"},{"Start":"04:16.355 ","End":"04:18.440","Text":"cosine of Alpha plus Beta,"},{"Start":"04:18.440 ","End":"04:26.990","Text":"which is cosine of 3x dx."},{"Start":"04:26.990 ","End":"04:29.490","Text":"We can just simplify."},{"Start":"04:31.010 ","End":"04:35.000","Text":"First of all, everything\u0027s going to be with 1/2."},{"Start":"04:35.000 ","End":"04:37.695","Text":"I can just take the 1/2 outside"},{"Start":"04:37.695 ","End":"04:42.365","Text":"the integral sign because the 1/2 is going to go on every term and it will take it out."},{"Start":"04:42.365 ","End":"04:49.930","Text":"What we have here is cosine 3x plus cosine of minus x."},{"Start":"04:49.930 ","End":"04:54.160","Text":"But cosine of minus x is the same as cosine of x."},{"Start":"04:54.160 ","End":"04:56.275","Text":"Let\u0027s write that at the side."},{"Start":"04:56.275 ","End":"05:00.910","Text":"It\u0027s well-known, cosine is an even function and cosine of minus x"},{"Start":"05:00.910 ","End":"05:05.405","Text":"is the same as cosine of x plus cosine of x."},{"Start":"05:05.405 ","End":"05:09.490","Text":"Then again, cosine of minus x,"},{"Start":"05:09.490 ","End":"05:14.055","Text":"which is the same as cosine of x."},{"Start":"05:14.055 ","End":"05:17.220","Text":"Finally, minus cosine of"},{"Start":"05:17.220 ","End":"05:26.050","Text":"3x, all this dx."},{"Start":"05:26.260 ","End":"05:28.600","Text":"Things are lucky for us."},{"Start":"05:28.600 ","End":"05:33.835","Text":"They simplify because the cosine 3x and the minus cosine 3x,"},{"Start":"05:33.835 ","End":"05:36.250","Text":"they cancel each other out."},{"Start":"05:36.250 ","End":"05:41.395","Text":"Inside the brackets, I\u0027m left with 2 cosine x."},{"Start":"05:41.395 ","End":"05:44.760","Text":"The 2 I can take outside with the 1/2."},{"Start":"05:44.760 ","End":"05:50.580","Text":"All I\u0027m left with is the integral of cosine x."},{"Start":"05:50.580 ","End":"05:53.324","Text":"Again, this last part, these 2 canceled,"},{"Start":"05:53.324 ","End":"05:57.225","Text":"these together, make 2 cosine x."},{"Start":"05:57.225 ","End":"06:00.940","Text":"The 2 with the 1/2 canceled,"},{"Start":"06:00.940 ","End":"06:03.985","Text":"and we just got the integral of cosine x dx."},{"Start":"06:03.985 ","End":"06:06.640","Text":"We know the integral of cosine x."},{"Start":"06:06.640 ","End":"06:09.980","Text":"This is just sine x plus"},{"Start":"06:09.980 ","End":"06:14.885","Text":"constant and that\u0027s the answer to this seemingly complicated problem."},{"Start":"06:14.885 ","End":"06:18.570","Text":"Next, we move on to part c."}],"ID":6717},{"Watched":false,"Name":"Exercise 5 Part c","Duration":"4m 39s","ChapterTopicVideoID":6658,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.617","Text":"Now, part C, which is here, is a tricky one,"},{"Start":"00:03.617 ","End":"00:07.500","Text":"and I\u0027m going to need several formulas which I\u0027ll write over here."},{"Start":"00:07.500 ","End":"00:09.570","Text":"We have all of these formulas."},{"Start":"00:09.570 ","End":"00:11.580","Text":"The first one straight from algebra,"},{"Start":"00:11.580 ","End":"00:14.115","Text":"you can check it by expanding here."},{"Start":"00:14.115 ","End":"00:17.535","Text":"Then 3 trigonometric formulas."},{"Start":"00:17.535 ","End":"00:21.510","Text":"The middle one here is not entirely standard."},{"Start":"00:21.510 ","End":"00:23.715","Text":"It\u0027s usually written with a 2 over here,"},{"Start":"00:23.715 ","End":"00:25.010","Text":"without the 1/2 over here,"},{"Start":"00:25.010 ","End":"00:26.355","Text":"but if we bring the 2 over,"},{"Start":"00:26.355 ","End":"00:27.810","Text":"we get it in this form."},{"Start":"00:27.810 ","End":"00:30.975","Text":"The last one\u0027s an integration formula."},{"Start":"00:30.975 ","End":"00:34.415","Text":"We\u0027ll begin with the algebra because this is what we have"},{"Start":"00:34.415 ","End":"00:38.200","Text":"with a equals sine x and b equals cosine x."},{"Start":"00:38.200 ","End":"00:44.028","Text":"What we get is the integral of"},{"Start":"00:44.028 ","End":"00:49.260","Text":"a^2 plus b^2 is sine^2 plus cosine^2,"},{"Start":"00:49.260 ","End":"00:52.472","Text":"all of this squared,"},{"Start":"00:52.472 ","End":"01:01.560","Text":"minus 2sine^2 x, cosine^2 x."},{"Start":"01:01.560 ","End":"01:03.960","Text":"Now, because of this formula,"},{"Start":"01:03.960 ","End":"01:05.835","Text":"with alpha replaced by x,"},{"Start":"01:05.835 ","End":"01:07.320","Text":"this thing is going to be 1,"},{"Start":"01:07.320 ","End":"01:08.400","Text":"and even if we square it,"},{"Start":"01:08.400 ","End":"01:10.260","Text":"it\u0027ll still stay 1."},{"Start":"01:10.260 ","End":"01:14.325","Text":"This is the integral of 1 minus,"},{"Start":"01:14.325 ","End":"01:18.143","Text":"now, I\u0027ll write the 2 here,"},{"Start":"01:18.143 ","End":"01:21.180","Text":"and the sine^2 x cosine^2 x,"},{"Start":"01:21.180 ","End":"01:26.475","Text":"I\u0027ll write as sine x cosine x,"},{"Start":"01:26.475 ","End":"01:29.655","Text":"all squared from the rules of exponents."},{"Start":"01:29.655 ","End":"01:32.950","Text":"I have been forgetting the dx."},{"Start":"01:33.080 ","End":"01:35.385","Text":"That\u0027s better,"},{"Start":"01:35.385 ","End":"01:40.155","Text":"and now we\u0027re going to use the formula for sine x cosine x from here,"},{"Start":"01:40.155 ","End":"01:44.145","Text":"sine alpha cosine alpha is 1/2 sine 2 alpha."},{"Start":"01:44.145 ","End":"01:48.160","Text":"The rest of it will stay the same, the 1 minus."},{"Start":"01:48.160 ","End":"01:54.380","Text":"Now, if sine x cosine x is 1/2 sine 2x,"},{"Start":"01:54.380 ","End":"02:02.385","Text":"when I square it, I\u0027m going to get 2 times 1/2 times 1/2 is 1/2."},{"Start":"02:02.385 ","End":"02:07.725","Text":"I\u0027m going to get for here, sine 2x, but also squared."},{"Start":"02:07.725 ","End":"02:15.165","Text":"It\u0027s sine^2 of 2x and all of this is dx."},{"Start":"02:15.165 ","End":"02:22.565","Text":"The next thing we\u0027re going to apply is this formula for sine^2 of alpha,"},{"Start":"02:22.565 ","End":"02:25.455","Text":"this time with alpha equaling 2x."},{"Start":"02:25.455 ","End":"02:34.595","Text":"What I get is the integral of 1 minus this part is minus 1/2."},{"Start":"02:34.595 ","End":"02:39.410","Text":"Using this formula for sine^2 alpha with alpha being 2x,"},{"Start":"02:39.410 ","End":"02:42.755","Text":"we get times another 1/2,"},{"Start":"02:42.755 ","End":"02:52.460","Text":"1 minus, since alpha is 2x, we get cosine 4x."},{"Start":"02:52.460 ","End":"02:54.590","Text":"Getting closer."},{"Start":"02:54.590 ","End":"02:58.730","Text":"Now at this point, just open brackets,"},{"Start":"02:58.730 ","End":"03:07.790","Text":"we\u0027ve got the integral of 1 minus 1/4 times 1,"},{"Start":"03:07.790 ","End":"03:10.700","Text":"which is minus 1/4,"},{"Start":"03:10.700 ","End":"03:17.310","Text":"then minus 1/4 times minus cosine 4x,"},{"Start":"03:17.310 ","End":"03:28.440","Text":"which is plus 1/4 cosine 4x, dx."},{"Start":"03:28.440 ","End":"03:32.085","Text":"1 minus 1/4 is 3/4."},{"Start":"03:32.085 ","End":"03:35.209","Text":"Just write this as the integral of"},{"Start":"03:35.209 ","End":"03:46.653","Text":"3/4 plus 1/4 cosine 4x, dx, which equals,"},{"Start":"03:46.653 ","End":"03:52.385","Text":"now just to sum and the constants multiplying, so they stay."},{"Start":"03:52.385 ","End":"03:55.865","Text":"What I get is the integral of 3/4"},{"Start":"03:55.865 ","End":"04:03.405","Text":"is 3/4 x plus 1/4 times the integral of cosine 4x."},{"Start":"04:03.405 ","End":"04:09.930","Text":"This I have here the integral of cosine ax is 1 over a times the sine."},{"Start":"04:09.930 ","End":"04:16.740","Text":"It\u0027s 1/4 times and then another 1/4, because it\u0027s 1 over this 4,"},{"Start":"04:16.740 ","End":"04:23.590","Text":"and sine of 4x plus a constant."},{"Start":"04:24.140 ","End":"04:30.245","Text":"That\u0027s basically it except that I would replace this by 1/16,"},{"Start":"04:30.245 ","End":"04:32.210","Text":"not just leave it like that,"},{"Start":"04:32.210 ","End":"04:34.655","Text":"but this is fine."},{"Start":"04:34.655 ","End":"04:36.470","Text":"That was a bit difficult,"},{"Start":"04:36.470 ","End":"04:40.410","Text":"but we\u0027re done with this exercise."}],"ID":6718},{"Watched":false,"Name":"Exercise 6","Duration":"7m 22s","ChapterTopicVideoID":6659,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.074","Text":"Here we have to compute the following 3 integrals."},{"Start":"00:03.074 ","End":"00:04.950","Text":"I\u0027ve already copied the first and"},{"Start":"00:04.950 ","End":"00:07.935","Text":"a couple of formulas we are going to need in solving it."},{"Start":"00:07.935 ","End":"00:12.600","Text":"The first thing to do here is to try and get rid of this cosine squared."},{"Start":"00:12.600 ","End":"00:14.100","Text":"It\u0027s something more like a sum or"},{"Start":"00:14.100 ","End":"00:17.310","Text":"a difference and we have exactly the formula for that here,"},{"Start":"00:17.310 ","End":"00:20.070","Text":"because if we replace Alpha with x,"},{"Start":"00:20.070 ","End":"00:22.810","Text":"we get 1 plus cosine 2 Alpha,"},{"Start":"00:22.810 ","End":"00:28.260","Text":"so over here we would get the half could come in front of the integral,"},{"Start":"00:28.260 ","End":"00:39.070","Text":"1/2 of the integral of 1 plus cosine 2x, dx."},{"Start":"00:39.070 ","End":"00:42.170","Text":"Now this 1 is immediate,"},{"Start":"00:42.170 ","End":"00:43.850","Text":"this 1 cosine 2x,"},{"Start":"00:43.850 ","End":"00:47.960","Text":"I have a formula for here, so it\u0027s 1/2."},{"Start":"00:47.960 ","End":"00:56.315","Text":"The integral of 1 is x and the integral of cosine 2x the 2 is this a."},{"Start":"00:56.315 ","End":"00:58.040","Text":"We have 1 over a,"},{"Start":"00:58.040 ","End":"01:06.935","Text":"which is 1 over 2 sine of 2x plus c and we could leave it like that,"},{"Start":"01:06.935 ","End":"01:12.230","Text":"or we could write it as 1/2x plus 1/4."},{"Start":"01:12.230 ","End":"01:20.765","Text":"That\u0027s from a half times a half sine 2x plus c. That\u0027s part a."},{"Start":"01:20.765 ","End":"01:25.340","Text":"Next. Well, I\u0027ve copied b over here."},{"Start":"01:25.340 ","End":"01:28.745","Text":"The formulas are going to have to change this one I can reuse."},{"Start":"01:28.745 ","End":"01:32.425","Text":"This one I am going to need the one for sine squared."},{"Start":"01:32.425 ","End":"01:39.905","Text":"I\u0027ll just erase this part here and make it the sine squared."},{"Start":"01:39.905 ","End":"01:43.400","Text":"The way the formula changes instead of plus,"},{"Start":"01:43.400 ","End":"01:45.020","Text":"I get a minus."},{"Start":"01:45.020 ","End":"01:52.924","Text":"Let\u0027s see if we can erase that plus and replace it with a minus."},{"Start":"01:52.924 ","End":"01:55.265","Text":"Now I have what I need."},{"Start":"01:55.265 ","End":"01:57.375","Text":"We\u0027ll use this formula first,"},{"Start":"01:57.375 ","End":"01:59.510","Text":"so it\u0027s equal to 1/2,"},{"Start":"01:59.510 ","End":"02:06.155","Text":"which will come in front of the integral sign of 1 minus cosine."},{"Start":"02:06.155 ","End":"02:08.630","Text":"Now, didn\u0027t say what Alpha is."},{"Start":"02:08.630 ","End":"02:11.195","Text":"But obviously I\u0027m letting Alpha going to be 4x,"},{"Start":"02:11.195 ","End":"02:17.590","Text":"so cosine 2 Alpha would be cosine of 8x, dx."},{"Start":"02:17.590 ","End":"02:20.020","Text":"Now this is equal to 1/2."},{"Start":"02:20.020 ","End":"02:26.555","Text":"The integral of 1 is immediate, that\u0027s x minus."},{"Start":"02:26.555 ","End":"02:32.690","Text":"Now the integral of cosine of 8x is exactly from this formula with a being 8."},{"Start":"02:32.690 ","End":"02:40.800","Text":"We have 1 over 8 sine of 8x plus c,"},{"Start":"02:40.800 ","End":"02:42.285","Text":"which will go at the end,"},{"Start":"02:42.285 ","End":"02:52.995","Text":"or I could write it as 1/2x minus 1/16 sine 8x plus c,"},{"Start":"02:52.995 ","End":"02:58.725","Text":"and so funny c, and that finishes part b."},{"Start":"02:58.725 ","End":"03:02.170","Text":"This is part c and I\u0027ve already"},{"Start":"03:02.170 ","End":"03:06.860","Text":"written a couple of trigonometric formulas that we\u0027re going to use."},{"Start":"03:06.860 ","End":"03:11.890","Text":"We have cosine cubed of x and the first thing that I\u0027ll"},{"Start":"03:11.890 ","End":"03:18.080","Text":"do is to write it as cosine times cosine squared."},{"Start":"03:18.080 ","End":"03:26.155","Text":"I have cosine of x times cosine squared of x."},{"Start":"03:26.155 ","End":"03:32.770","Text":"Now I can use the formula for cosine squared of x from here and I will get that"},{"Start":"03:32.770 ","End":"03:39.880","Text":"this equals the integral of cosine x times 1/2,"},{"Start":"03:39.880 ","End":"03:41.665","Text":"which I put here,"},{"Start":"03:41.665 ","End":"03:47.855","Text":"times 1 plus cosine of 2x,"},{"Start":"03:47.855 ","End":"03:51.425","Text":"dx here I\u0027m replacing Alpha by x."},{"Start":"03:51.425 ","End":"03:57.280","Text":"Next, open the brackets and get 1/2 of the integral of cosine x times"},{"Start":"03:57.280 ","End":"04:03.680","Text":"1 is cosine x plus cosine x cosine 2x."},{"Start":"04:06.510 ","End":"04:17.250","Text":"Maybe the brackets will make it clearer and all of this dx."},{"Start":"04:17.250 ","End":"04:23.145","Text":"What I\u0027ll do, is I\u0027ll leave the first term as it is cosine x,"},{"Start":"04:23.145 ","End":"04:30.340","Text":"but this, I\u0027ll use this formula with Alpha being x and Beta being 2x."},{"Start":"04:30.340 ","End":"04:37.945","Text":"I get plus 1/2 of cosine of the sum,"},{"Start":"04:37.945 ","End":"04:43.620","Text":"which is cosine 3x plus cosine of the difference."},{"Start":"04:43.620 ","End":"04:47.890","Text":"The difference is x minus 2x is minus x,"},{"Start":"04:47.890 ","End":"04:51.430","Text":"but cosine of minus x and cosine x are the same."},{"Start":"04:51.430 ","End":"04:52.540","Text":"It\u0027s an even function,"},{"Start":"04:52.540 ","End":"04:54.180","Text":"so I\u0027m just indicating it,"},{"Start":"04:54.180 ","End":"04:55.560","Text":"really should have been minus x,"},{"Start":"04:55.560 ","End":"05:00.719","Text":"but it\u0027s the same, plus cosine of that, dx."},{"Start":"05:00.719 ","End":"05:03.705","Text":"Just open the brackets,"},{"Start":"05:03.705 ","End":"05:13.340","Text":"1/2 of the integral of cosine x plus 1/2 cosine 3x from here,"},{"Start":"05:13.340 ","End":"05:21.450","Text":"plus 1/2 cosine of x from here, all this dx."},{"Start":"05:21.450 ","End":"05:26.750","Text":"Altogether, if I collect cosine x\u0027s, I have 1/2,"},{"Start":"05:26.750 ","End":"05:31.950","Text":"so I have 3 over 2 cosine x"},{"Start":"05:31.950 ","End":"05:37.380","Text":"plus 1/2 cosine of 3x."},{"Start":"05:37.380 ","End":"05:42.140","Text":"At this point, maybe I\u0027ll bring in another formula and that is that the"},{"Start":"05:42.140 ","End":"05:47.510","Text":"integral of cosine of ax instead of"},{"Start":"05:47.510 ","End":"05:52.970","Text":"x is equal to 1 over a times"},{"Start":"05:52.970 ","End":"05:58.860","Text":"the sine of ax plus c. If I use that here,"},{"Start":"05:58.860 ","End":"06:02.149","Text":"1 time with a equals 1,"},{"Start":"06:02.149 ","End":"06:03.970","Text":"well then I don\u0027t really need it,"},{"Start":"06:03.970 ","End":"06:07.340","Text":"but for here, for it with a equals 3."},{"Start":"06:07.490 ","End":"06:14.020","Text":"We have 1/2. Now, this thing is an immediate integral because for cosine x,"},{"Start":"06:14.020 ","End":"06:16.930","Text":"we know it\u0027s just sine x. I don\u0027t need this generalization,"},{"Start":"06:16.930 ","End":"06:20.415","Text":"so I\u0027ve got 3 over 2 sine x,"},{"Start":"06:20.415 ","End":"06:22.305","Text":"but for the second part,"},{"Start":"06:22.305 ","End":"06:29.680","Text":"for the cosine 3x I need to use this formula with a equals 3 and I get 1/2."},{"Start":"06:29.680 ","End":"06:32.290","Text":"What did I write here? I\u0027m sorry."},{"Start":"06:32.290 ","End":"06:35.575","Text":"I meant to write 1 over a, excuse me."},{"Start":"06:35.575 ","End":"06:37.915","Text":"My apologies. The half was here."},{"Start":"06:37.915 ","End":"06:41.195","Text":"The 3 gives me 1 over 3,"},{"Start":"06:41.195 ","End":"06:47.840","Text":"so 1 over 3 sine of 3x plus the c. If we want to,"},{"Start":"06:47.840 ","End":"06:49.790","Text":"we can open bracket, let\u0027s see,"},{"Start":"06:49.790 ","End":"06:54.095","Text":"1/2 times 3/2 is just 3/4,"},{"Start":"06:54.095 ","End":"07:00.335","Text":"so we have 3/4 of sine x and 1/2 times 1/3 is 1/6,"},{"Start":"07:00.335 ","End":"07:06.805","Text":"plus 1/6 sine 3x plus the constant."},{"Start":"07:06.805 ","End":"07:08.690","Text":"I just caught a mistake."},{"Start":"07:08.690 ","End":"07:13.610","Text":"I forgot this 1/2 here when I multiplied the 1/2 times the 1/3 so it\u0027s not 1/6,"},{"Start":"07:13.610 ","End":"07:17.510","Text":"it is actually 1 over 12."},{"Start":"07:17.510 ","End":"07:22.740","Text":"Apologies. Anyway, that does part c and we\u0027re finished here."}],"ID":6719},{"Watched":false,"Name":"Exercise 7 Parts a-b","Duration":"9m 39s","ChapterTopicVideoID":6660,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.130","Text":"In this exercise, we have to compute the following integrals. There are 3 of them."},{"Start":"00:06.290 ","End":"00:09.120","Text":"I\u0027ve written out the first one,"},{"Start":"00:09.120 ","End":"00:14.545","Text":"and I\u0027ve also written the formulas we\u0027re going to need while solving it."},{"Start":"00:14.545 ","End":"00:20.125","Text":"The first thing to do is some algebra to break down this power of 3."},{"Start":"00:20.125 ","End":"00:23.030","Text":"Since we have a formula for the power of 2,"},{"Start":"00:23.030 ","End":"00:29.180","Text":"let\u0027s write it as the integral of sine times sine squared."},{"Start":"00:29.180 ","End":"00:33.575","Text":"We have sine 4x times"},{"Start":"00:33.575 ","End":"00:39.570","Text":"sine squared 4x dx."},{"Start":"00:39.570 ","End":"00:43.505","Text":"Now we can use the formula for sine squared,"},{"Start":"00:43.505 ","End":"00:50.880","Text":"which is this, and I\u0027ll put the 1/2 in front of the integral sine."},{"Start":"00:51.680 ","End":"00:55.730","Text":"I\u0027m going to expand this using Alpha equals"},{"Start":"00:55.730 ","End":"01:00.740","Text":"4x and I\u0027m putting the 1/2 in front of the integral sine,"},{"Start":"01:00.740 ","End":"01:06.215","Text":"so I get 1/2 the integral of sine 4x."},{"Start":"01:06.215 ","End":"01:14.140","Text":"Then from here, 1 minus cosine 2 Alpha will be 8x dx."},{"Start":"01:15.770 ","End":"01:22.250","Text":"We get 1/2 of the integral of multiplying out"},{"Start":"01:22.250 ","End":"01:27.790","Text":"sine 4x minus sine"},{"Start":"01:27.790 ","End":"01:34.350","Text":"4x cosine 8x."},{"Start":"01:34.350 ","End":"01:39.180","Text":"Now, you see why I wrote the formula for sine Alpha cosine Beta."},{"Start":"01:39.220 ","End":"01:47.310","Text":"This will equal 1/2 integral of sine 4x minus,"},{"Start":"01:47.310 ","End":"01:53.355","Text":"let\u0027s see, sine 4x cosine 8x means Alpha is 4x, Beta is 8x."},{"Start":"01:53.355 ","End":"01:55.810","Text":"We get sine of Alpha plus Beta,"},{"Start":"01:55.810 ","End":"01:57.624","Text":"but this is all within a minus,"},{"Start":"01:57.624 ","End":"02:00.185","Text":"and that will also be a minus."},{"Start":"02:00.185 ","End":"02:02.355","Text":"But there\u0027s also 1/2,"},{"Start":"02:02.355 ","End":"02:05.160","Text":"so it\u0027s minus 1/2 sine of"},{"Start":"02:05.160 ","End":"02:13.090","Text":"12x and minus 1/2 sine of the difference,"},{"Start":"02:13.090 ","End":"02:24.590","Text":"sine of minus 4x which is 4x minus 8x dx."},{"Start":"02:24.590 ","End":"02:27.545","Text":"We get, let\u0027s see now,"},{"Start":"02:27.545 ","End":"02:30.380","Text":"this minus with this minus goes because sine is"},{"Start":"02:30.380 ","End":"02:33.545","Text":"an odd function and the minus comes outside."},{"Start":"02:33.545 ","End":"02:37.860","Text":"We get plus 1 plus 1/2 sine 4x."},{"Start":"02:37.890 ","End":"02:40.840","Text":"We get 3 over 2"},{"Start":"02:40.840 ","End":"02:49.205","Text":"sine 4x minus 1/2 sine"},{"Start":"02:49.205 ","End":"02:55.290","Text":"12x dx."},{"Start":"02:55.290 ","End":"03:04.935","Text":"I\u0027ll just write the integral of sine of ax dx is equal to 1 over a,"},{"Start":"03:04.935 ","End":"03:08.430","Text":"but it\u0027s minus cosine so I\u0027ll put the minus here,"},{"Start":"03:08.430 ","End":"03:17.450","Text":"cosine of ax plus c. When there\u0027s an a in front of the x,"},{"Start":"03:17.450 ","End":"03:20.405","Text":"then we have to divide by a."},{"Start":"03:20.405 ","End":"03:21.635","Text":"What we would normally do,"},{"Start":"03:21.635 ","End":"03:26.485","Text":"integrals of sine is normally minus cosine but if it\u0027s an a, it\u0027s divided by."},{"Start":"03:26.485 ","End":"03:33.140","Text":"In our case, we\u0027re going to let a at one time be 4 and at one time be 12."},{"Start":"03:33.170 ","End":"03:38.025","Text":"So we get, let\u0027s see,"},{"Start":"03:38.025 ","End":"03:43.845","Text":"the integral of sine 4x is going to be minus 1/4 cosine 4x."},{"Start":"03:43.845 ","End":"03:46.320","Text":"We have minus 1/4,"},{"Start":"03:46.320 ","End":"03:47.930","Text":"let me just do this at the side."},{"Start":"03:47.930 ","End":"03:53.165","Text":"We have 1/2 times 3 over 2 times minus 1/4."},{"Start":"03:53.165 ","End":"03:56.510","Text":"I make this minus 3 over 16."},{"Start":"03:56.510 ","End":"04:06.300","Text":"We have minus 3 over 16 times the cosine of 4x."},{"Start":"04:06.300 ","End":"04:10.505","Text":"Then the minus with the minus will become a plus."},{"Start":"04:10.505 ","End":"04:13.010","Text":"We\u0027ll get minus 1 over 12."},{"Start":"04:13.010 ","End":"04:16.970","Text":"For this part, minus 1 over 12 cosine x."},{"Start":"04:16.970 ","End":"04:25.150","Text":"The minus 1/2 with the minus 1 over 12 will give me 1 over 24."},{"Start":"04:25.150 ","End":"04:31.470","Text":"It\u0027s plus 1 over 24 cosine of 12x"},{"Start":"04:31.470 ","End":"04:38.870","Text":"and plus C. Just caught it in time."},{"Start":"04:38.870 ","End":"04:42.620","Text":"This is an extra 1/2 here before the minus 1/2."},{"Start":"04:42.620 ","End":"04:50.865","Text":"There\u0027s a 1/2 here times this that brings this instead to 1 over 48."},{"Start":"04:50.865 ","End":"04:56.325","Text":"I will erase the 24 and in its place write 48."},{"Start":"04:56.325 ","End":"05:00.760","Text":"That looks better to me now. Done with part a."},{"Start":"05:00.890 ","End":"05:04.935","Text":"Here\u0027s part b, cosine to the fourth."},{"Start":"05:04.935 ","End":"05:08.165","Text":"The formula we\u0027re going to need from trigonometry,"},{"Start":"05:08.165 ","End":"05:10.635","Text":"cosine squared Alpha equals 1/2,"},{"Start":"05:10.635 ","End":"05:13.440","Text":"1 plus cosine of 2 Alpha."},{"Start":"05:13.440 ","End":"05:18.780","Text":"Let\u0027s use this right away with Alpha being x here."},{"Start":"05:18.780 ","End":"05:21.645","Text":"What we have is the integral."},{"Start":"05:21.645 ","End":"05:25.854","Text":"Now, cosine to the fourth is cosine squared squared."},{"Start":"05:25.854 ","End":"05:29.830","Text":"Now, the cosine squared of x is 1/2,"},{"Start":"05:29.830 ","End":"05:37.115","Text":"1 plus cosine 2x."},{"Start":"05:37.115 ","End":"05:40.195","Text":"That part is the cosine squared of x"},{"Start":"05:40.195 ","End":"05:44.180","Text":"and make it to the power of fourth all this is squared."},{"Start":"05:44.390 ","End":"05:47.710","Text":"Now, we can square this thing."},{"Start":"05:47.710 ","End":"05:52.884","Text":"First of all, the 1/2 squared is 1/4 and that can come out of the integral sign."},{"Start":"05:52.884 ","End":"05:56.465","Text":"We have 1 plus cosine 2x all squared."},{"Start":"05:56.465 ","End":"05:59.020","Text":"Using the a plus b squared formula,"},{"Start":"05:59.020 ","End":"06:04.030","Text":"it\u0027s 1 squared is 1 plus twice 1 times cosine 2x,"},{"Start":"06:04.030 ","End":"06:10.225","Text":"which is twice cosine 2x plus this thing squared,"},{"Start":"06:10.225 ","End":"06:19.640","Text":"which is cosine squared 2x and all this dx."},{"Start":"06:21.290 ","End":"06:25.950","Text":"Everything here is straightforward except for the last one,"},{"Start":"06:25.950 ","End":"06:29.110","Text":"which also will be straightforward once I\u0027ve shown you,"},{"Start":"06:29.110 ","End":"06:33.315","Text":"which is 1 plus 2 cosine 2x."},{"Start":"06:33.315 ","End":"06:34.910","Text":"Now, this cosine squared,"},{"Start":"06:34.910 ","End":"06:37.325","Text":"I can use this formula again,"},{"Start":"06:37.325 ","End":"06:40.310","Text":"this time with Alpha being 2x."},{"Start":"06:40.310 ","End":"06:47.000","Text":"Here I have plus 1/2,1"},{"Start":"06:47.000 ","End":"06:55.700","Text":"plus cosine 4x dx."},{"Start":"06:55.700 ","End":"07:01.095","Text":"Let\u0027s expand and we get 1/4 of the integral."},{"Start":"07:01.095 ","End":"07:03.569","Text":"Now, from here we get 1,"},{"Start":"07:03.569 ","End":"07:05.655","Text":"and from here we get 1/2."},{"Start":"07:05.655 ","End":"07:10.485","Text":"That\u0027s 3 over 2 plus"},{"Start":"07:10.485 ","End":"07:18.070","Text":"twice cosine 2x and plus 1/2 cosine 4x,"},{"Start":"07:19.010 ","End":"07:25.695","Text":"1/2 cosine 4x dx."},{"Start":"07:25.695 ","End":"07:32.605","Text":"Now, let\u0027s remember another formula that the integral of the cosine of"},{"Start":"07:32.605 ","End":"07:42.360","Text":"ax or ax plus b dx is not simply the sine of ax,"},{"Start":"07:42.360 ","End":"07:44.415","Text":"as you might think,"},{"Start":"07:44.415 ","End":"07:54.140","Text":"but it\u0027s 1 over a of this plus c. Here we\u0027re going to use it twice,"},{"Start":"07:54.140 ","End":"07:58.940","Text":"once with a equals 2 here and once with the a equals 4,"},{"Start":"07:58.940 ","End":"08:03.485","Text":"maybe I\u0027ll use a bit of highlighting just to say if we have ax here,"},{"Start":"08:03.485 ","End":"08:05.620","Text":"then we divide by a,"},{"Start":"08:05.620 ","End":"08:08.500","Text":"and here we have a is 2,"},{"Start":"08:08.500 ","End":"08:10.760","Text":"and here we have a is 4."},{"Start":"08:10.760 ","End":"08:13.235","Text":"So that will help us."},{"Start":"08:13.235 ","End":"08:18.510","Text":"What we get is I\u0027ll keep the"},{"Start":"08:18.510 ","End":"08:23.225","Text":"1/4 here and we\u0027ll have each of these integrals added separately."},{"Start":"08:23.225 ","End":"08:24.530","Text":"I\u0027ll just put this in a bracket,"},{"Start":"08:24.530 ","End":"08:28.090","Text":"it\u0027s not constant, it\u0027s just a constant times x."},{"Start":"08:28.090 ","End":"08:32.790","Text":"This cosine 2x is 2 times, now,"},{"Start":"08:32.790 ","End":"08:37.660","Text":"from this formula, it\u0027s 1/2 sine of 2x."},{"Start":"08:37.660 ","End":"08:40.385","Text":"See we write one time at the end."},{"Start":"08:40.385 ","End":"08:47.165","Text":"Here we have 1/2 times 1/4 sine"},{"Start":"08:47.165 ","End":"08:54.425","Text":"4x plus c. Let\u0027s see what we get in the end."},{"Start":"08:54.425 ","End":"08:56.720","Text":"We get 1/4 times 3 over 2,"},{"Start":"08:56.720 ","End":"09:00.680","Text":"which is 3/8, that\u0027s the number,"},{"Start":"09:00.680 ","End":"09:05.625","Text":"plus 2 times 1/2 is 1 plus sine of 2x,"},{"Start":"09:05.625 ","End":"09:13.955","Text":"and 1/8 of sine 4x plus a constant."},{"Start":"09:13.955 ","End":"09:17.960","Text":"Must excuse me, I forgot about the 1/4 in the last 2 terms,"},{"Start":"09:17.960 ","End":"09:19.310","Text":"so let me factor those in."},{"Start":"09:19.310 ","End":"09:21.740","Text":"So this 1/4 I\u0027ll put in here as"},{"Start":"09:21.740 ","End":"09:26.960","Text":"1/4 and I\u0027ll fit it in here as 1/4 times 1/8,"},{"Start":"09:26.960 ","End":"09:30.695","Text":"which is 1 over 32 instead of the 8,"},{"Start":"09:30.695 ","End":"09:33.229","Text":"just forgive me about that."},{"Start":"09:33.229 ","End":"09:35.975","Text":"This I believe should be correct."},{"Start":"09:35.975 ","End":"09:39.210","Text":"We\u0027re done with part b."}],"ID":6720},{"Watched":false,"Name":"Exercise 7 Part c","Duration":"4m 18s","ChapterTopicVideoID":6661,"CourseChapterTopicPlaylistID":1608,"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":"I copied Part C over here."},{"Start":"00:03.315 ","End":"00:06.300","Text":"I kept the formulas from Part B,"},{"Start":"00:06.300 ","End":"00:10.190","Text":"which is actually quite similar to C. But I also added another one;"},{"Start":"00:10.190 ","End":"00:12.750","Text":"the cosine squared is not enough anymore,"},{"Start":"00:12.750 ","End":"00:16.320","Text":"I also need a sine squared of Alpha,"},{"Start":"00:16.320 ","End":"00:18.705","Text":"and that\u0027s what we\u0027re going to open with."},{"Start":"00:18.705 ","End":"00:25.155","Text":"Because here I can see this fourth power as sine squared all squared."},{"Start":"00:25.155 ","End":"00:27.310","Text":"I have the integral."},{"Start":"00:27.310 ","End":"00:34.010","Text":"If I expand sine squared in here and then make it squared dx."},{"Start":"00:34.010 ","End":"00:38.830","Text":"Now if I use this formula with Alpha being 2x,"},{"Start":"00:38.830 ","End":"00:47.235","Text":"I will get that the sine squared of Alpha is 1/2 of 1 minus cosine,"},{"Start":"00:47.235 ","End":"00:52.030","Text":"and 2 Alpha, if Alpha is 2x, is 4x."},{"Start":"00:52.790 ","End":"00:56.265","Text":"1/2 squared is a quarter,"},{"Start":"00:56.265 ","End":"00:59.825","Text":"and it\u0027s a constant, so it can come out of the integral sign."},{"Start":"00:59.825 ","End":"01:04.160","Text":"What I\u0027m left with is the 1 minus cosine 4x all squared,"},{"Start":"01:04.160 ","End":"01:13.725","Text":"which is 1 minus twice cosine 4x plus cosine squared 4x,"},{"Start":"01:13.725 ","End":"01:19.875","Text":"from the standard a minus b all squared in the algebraic formula."},{"Start":"01:19.875 ","End":"01:22.770","Text":"These are all becoming straightforward, these terms."},{"Start":"01:22.770 ","End":"01:24.280","Text":"The last 1, the cosine squared,"},{"Start":"01:24.280 ","End":"01:25.720","Text":"I want to get rid of."},{"Start":"01:25.720 ","End":"01:29.695","Text":"But I kept from last time the cosine squared Alpha formula,"},{"Start":"01:29.695 ","End":"01:33.860","Text":"so I can now use it, and I get 1/4,"},{"Start":"01:33.860 ","End":"01:41.155","Text":"the integral of 1 minus 2 cosine 4x plus,"},{"Start":"01:41.155 ","End":"01:43.720","Text":"here Alpha is going to be 4x,"},{"Start":"01:43.720 ","End":"01:48.565","Text":"so I get 1/2 of 1"},{"Start":"01:48.565 ","End":"01:57.090","Text":"plus cosine 8x dx."},{"Start":"01:57.090 ","End":"02:00.339","Text":"If I tidy up here a bit,"},{"Start":"02:00.710 ","End":"02:04.935","Text":"just to collect like terms together, constants,"},{"Start":"02:04.935 ","End":"02:11.115","Text":"I have 1 plus 1/2 is 1 and 1/2,"},{"Start":"02:11.115 ","End":"02:15.730","Text":"3 over 2 times a 1/4 is 3/8."},{"Start":"02:15.730 ","End":"02:19.670","Text":"Let\u0027s see, how many times cosine 4x do I have."},{"Start":"02:19.670 ","End":"02:26.335","Text":"The minus 2 over the 4 is minus 1/2 of the cosine 4x."},{"Start":"02:26.335 ","End":"02:31.250","Text":"Finally, the cosine 8x,"},{"Start":"02:31.250 ","End":"02:33.940","Text":"I have 1/2 times 1/4,"},{"Start":"02:33.940 ","End":"02:36.535","Text":"which is 1/8 of 1 of these,"},{"Start":"02:36.535 ","End":"02:39.620","Text":"of a cosine 8x dx."},{"Start":"02:40.930 ","End":"02:45.515","Text":"Now I\u0027ll take the integral of each of these 3 pieces separately,"},{"Start":"02:45.515 ","End":"02:50.120","Text":"and I\u0027ll also be using this formula twice,"},{"Start":"02:50.120 ","End":"02:56.910","Text":"once with the a being 4 and once with a being 8."},{"Start":"02:58.430 ","End":"03:01.400","Text":"Let\u0027s see what I get."},{"Start":"03:01.400 ","End":"03:07.925","Text":"I will get the integral of 3/8 is 3/8x."},{"Start":"03:07.925 ","End":"03:15.315","Text":"The integral of cosine is minus the 1 over a,"},{"Start":"03:15.315 ","End":"03:17.629","Text":"but the minus goes with the minus."},{"Start":"03:17.629 ","End":"03:23.105","Text":"In other words, I\u0027m going to get minus 1/4 times a minus 1/2,"},{"Start":"03:23.105 ","End":"03:30.060","Text":"minus 1/4, it\u0027s just 1 over a, which is 1 over 4."},{"Start":"03:30.060 ","End":"03:31.695","Text":"So it\u0027s going to be here,"},{"Start":"03:31.695 ","End":"03:37.510","Text":"minus 1/2 times 1/4 minus 1/8,"},{"Start":"03:37.610 ","End":"03:45.090","Text":"and it\u0027s going to be sine of 4x."},{"Start":"03:45.090 ","End":"03:47.405","Text":"Now the integral of this."},{"Start":"03:47.405 ","End":"03:51.900","Text":"The integral of cosine once again,"},{"Start":"03:51.900 ","End":"03:55.785","Text":"it\u0027s going to be 1 over 8, but the 1 over 8,"},{"Start":"03:55.785 ","End":"03:59.160","Text":"there already is a 1 over 8 here,"},{"Start":"03:59.160 ","End":"04:09.460","Text":"so we get 1 over 64 times 8 of sine of 8x,"},{"Start":"04:09.500 ","End":"04:15.570","Text":"and finally plus C. This concludes the last of the set of 3,"},{"Start":"04:15.570 ","End":"04:18.600","Text":"and so we\u0027re done with this exercise."}],"ID":6721},{"Watched":false,"Name":"Exercise 8","Duration":"7m 23s","ChapterTopicVideoID":6662,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have to compute the following integrals, A,"},{"Start":"00:03.510 ","End":"00:05.820","Text":"B, and C. Start with A,"},{"Start":"00:05.820 ","End":"00:07.545","Text":"I\u0027ve copied it here."},{"Start":"00:07.545 ","End":"00:12.540","Text":"Now I\u0027ve written down a pair of formulas that are really going to help us here,"},{"Start":"00:12.540 ","End":"00:14.820","Text":"these are the well-known cosine squared and sine"},{"Start":"00:14.820 ","End":"00:19.525","Text":"squared in terms of cosine twice the angle."},{"Start":"00:19.525 ","End":"00:22.730","Text":"I suppose I should have written it with Alpha."},{"Start":"00:22.730 ","End":"00:26.975","Text":"What I have here is very similar to what I have on the right-hand side,"},{"Start":"00:26.975 ","End":"00:30.050","Text":"here and here, except that without the 1/2."},{"Start":"00:30.050 ","End":"00:32.275","Text":"We can easily fix that."},{"Start":"00:32.275 ","End":"00:38.560","Text":"I just have to put in a 1/2 here and 1/2 here."},{"Start":"00:38.560 ","End":"00:40.780","Text":"If I do it to the top and bottom, That\u0027s okay."},{"Start":"00:40.780 ","End":"00:43.340","Text":"Now I have exactly the right-hand sides."},{"Start":"00:43.340 ","End":"00:46.880","Text":"This will equal the left-hand sides."},{"Start":"00:46.880 ","End":"00:50.970","Text":"Integral of cosine squared X,"},{"Start":"00:50.970 ","End":"00:52.835","Text":"where Alpha is X here,"},{"Start":"00:52.835 ","End":"01:00.430","Text":"and over sine squared of X, dx."},{"Start":"01:01.030 ","End":"01:04.730","Text":"I can use another formula that"},{"Start":"01:04.730 ","End":"01:13.345","Text":"cosine squared Alpha plus sine squared Alpha is equal to 1."},{"Start":"01:13.345 ","End":"01:16.520","Text":"Instead of cosine squared Alpha,"},{"Start":"01:16.520 ","End":"01:24.920","Text":"I can write 1 minus sine squared Alpha over sine squared Alpha."},{"Start":"01:24.920 ","End":"01:27.005","Text":"I\u0027m using X over here."},{"Start":"01:27.005 ","End":"01:30.070","Text":"Fix that in a second. There we go."},{"Start":"01:30.070 ","End":"01:32.360","Text":"Now using distributive law,"},{"Start":"01:32.360 ","End":"01:37.285","Text":"it\u0027s 1 over sine squared X and this over this,"},{"Start":"01:37.285 ","End":"01:44.260","Text":"it\u0027s the integral of 1 over sine squared X."},{"Start":"01:45.140 ","End":"01:49.840","Text":"This over this minus this over this is 1."},{"Start":"01:49.840 ","End":"01:52.480","Text":"Sine squared over sine squared is just 1."},{"Start":"01:52.480 ","End":"01:54.725","Text":"This is what I get,"},{"Start":"01:54.725 ","End":"01:58.375","Text":"now I can do the integral of each ones."},{"Start":"01:58.375 ","End":"02:03.640","Text":"What we get is an immediate integral because the integral of"},{"Start":"02:03.640 ","End":"02:11.000","Text":"1 over sine squared is minus cotangent,"},{"Start":"02:11.000 ","End":"02:18.330","Text":"the integral of minus 1 is just minus X and plus a"},{"Start":"02:18.330 ","End":"02:27.510","Text":"constant that\u0027s it for Part A. I copied part B over here."},{"Start":"02:27.510 ","End":"02:31.444","Text":"We see both on the numerator and the denominator,"},{"Start":"02:31.444 ","End":"02:33.640","Text":"we have sine minus sine."},{"Start":"02:33.640 ","End":"02:38.930","Text":"I brought in this formula for sine Alpha minus sine Beta over here."},{"Start":"02:38.930 ","End":"02:42.425","Text":"I know later we\u0027re going to need the sine 2 Alpha formula,"},{"Start":"02:42.425 ","End":"02:44.345","Text":"I wrote it down already."},{"Start":"02:44.345 ","End":"02:48.380","Text":"But meanwhile, let\u0027s get started with opening it"},{"Start":"02:48.380 ","End":"02:53.090","Text":"up according to the sine Alpha minus sine Beta formula."},{"Start":"02:53.090 ","End":"02:56.825","Text":"Here we have 5x and X for Alpha-Beta."},{"Start":"02:56.825 ","End":"03:06.720","Text":"We have on the numerator 2 sine of 1/2 the difference 5x minus X over"},{"Start":"03:06.720 ","End":"03:11.280","Text":"2 is 2x cosine"},{"Start":"03:11.280 ","End":"03:18.090","Text":"of 1/2 the sum 5x plus X over 2 is 3x."},{"Start":"03:18.090 ","End":"03:26.985","Text":"Then on the denominator we have twice sine 1/2 the difference."},{"Start":"03:26.985 ","End":"03:29.610","Text":"This time it\u0027s 4 minus 2 over 2."},{"Start":"03:29.610 ","End":"03:35.940","Text":"That\u0027s just sine X and cosine of 1/2 the sum,"},{"Start":"03:35.940 ","End":"03:42.210","Text":"that\u0027s cosine 4x plus 2x over 2 is 3x."},{"Start":"03:42.210 ","End":"03:49.305","Text":"What luck? Because, first of all, the 2 is canceled and then the cosine 3x cancels."},{"Start":"03:49.305 ","End":"03:53.775","Text":"All we\u0027re left with is sine 2x over sine X."},{"Start":"03:53.775 ","End":"03:57.430","Text":"At this point, instead of sine 2x."},{"Start":"03:57.430 ","End":"04:04.795","Text":"We use this formula with Alpha equals X to get 2 sine X cosine X."},{"Start":"04:04.795 ","End":"04:11.150","Text":"On the bottom sine X and we continue to be lucky,"},{"Start":"04:11.150 ","End":"04:13.025","Text":"this cancels with this."},{"Start":"04:13.025 ","End":"04:18.915","Text":"All we have is twice the integral of cosine X. Take the 2 out."},{"Start":"04:18.915 ","End":"04:21.060","Text":"Integral of cosine X,"},{"Start":"04:21.060 ","End":"04:24.135","Text":"dx is just sine X."},{"Start":"04:24.135 ","End":"04:29.560","Text":"The answer is 2 sine X plus a constant."},{"Start":"04:29.570 ","End":"04:33.295","Text":"That\u0027s it for Part B."},{"Start":"04:33.295 ","End":"04:36.940","Text":"Well, here I\u0027ve copied part C,"},{"Start":"04:36.940 ","End":"04:40.400","Text":"these are the formulas I\u0027m going to need. This 1 I kept from before."},{"Start":"04:42.260 ","End":"04:46.030","Text":"There\u0027s a pair of famous formulas accepted."},{"Start":"04:46.030 ","End":"04:52.160","Text":"They\u0027re sometimes known slightly differently with the 2 over here as 1/2."},{"Start":"04:52.160 ","End":"04:56.530","Text":"You may have seen sine squared X is 1/2 of 1 minus cosine 2x."},{"Start":"04:56.530 ","End":"04:59.455","Text":"Anyway, the 2 is sometimes on the left sometimes on the right."},{"Start":"04:59.455 ","End":"05:04.845","Text":"These are the formulas and we\u0027ll use these 2 with Alpha."},{"Start":"05:04.845 ","End":"05:07.295","Text":"I should have written Alpha instead of X,"},{"Start":"05:07.295 ","End":"05:10.570","Text":"there, none that there\u0027s wrong with the X but,"},{"Start":"05:10.570 ","End":"05:13.370","Text":"trig formulas are usually with Alpha."},{"Start":"05:13.860 ","End":"05:20.670","Text":"What we get here is from the sine 2x with Alpha equals X,"},{"Start":"05:20.670 ","End":"05:27.000","Text":"I get 2 sine X, cosine X."},{"Start":"05:27.000 ","End":"05:31.020","Text":"Likewise on the denominator,"},{"Start":"05:31.020 ","End":"05:37.405","Text":"sine 2x is 2 sine X, cosine X."},{"Start":"05:37.405 ","End":"05:42.370","Text":"Now what I have here, if I just change the order is 1 minus cosine 2x."},{"Start":"05:42.370 ","End":"05:47.865","Text":"We\u0027ll have 1 minus cosine 2x, it\u0027s 2 sine squared X."},{"Start":"05:47.865 ","End":"05:55.290","Text":"Here we have plus 2 sine squared X."},{"Start":"05:55.290 ","End":"05:58.560","Text":"Here we have 1 plus cosine 2x,"},{"Start":"05:58.560 ","End":"06:01.335","Text":"it\u0027s 2, cosine squared X."},{"Start":"06:01.335 ","End":"06:10.760","Text":"Okay? Continuing, I can take out of the brackets in the numerator 2 sine X."},{"Start":"06:10.760 ","End":"06:16.500","Text":"What we have is integral of 2 sine X."},{"Start":"06:16.500 ","End":"06:23.595","Text":"Here in brackets, I have cosine X plus sine X over."},{"Start":"06:23.595 ","End":"06:27.420","Text":"Here, I can take 2 cosine X outside the brackets"},{"Start":"06:27.420 ","End":"06:33.550","Text":"2 cosine X. I also get sine X plus cosine X."},{"Start":"06:33.940 ","End":"06:36.530","Text":"Okay, in reverse order."},{"Start":"06:36.530 ","End":"06:38.210","Text":"But still it\u0027s the same thing,"},{"Start":"06:38.210 ","End":"06:40.985","Text":"which means that this can cancel with this."},{"Start":"06:40.985 ","End":"06:42.350","Text":"As a matter of fact, the 2,"},{"Start":"06:42.350 ","End":"06:43.910","Text":"will go with the 2."},{"Start":"06:43.910 ","End":"06:47.960","Text":"All we\u0027re left with is the integral of sine over cosine,"},{"Start":"06:47.960 ","End":"06:55.410","Text":"which is tangent X, dx."},{"Start":"06:55.410 ","End":"07:00.840","Text":"There\u0027s a formula for that which says that it\u0027s equal to."}],"ID":6722},{"Watched":false,"Name":"Exercise 9","Duration":"23m 50s","ChapterTopicVideoID":1670,"CourseChapterTopicPlaylistID":1608,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.610","Text":"In this exercise, we have to compute the following integrals, there\u0027s 3 of them."},{"Start":"00:05.610 ","End":"00:07.140","Text":"Let\u0027s start with the first,"},{"Start":"00:07.140 ","End":"00:09.060","Text":"which I\u0027ve already copied over here."},{"Start":"00:09.060 ","End":"00:13.440","Text":"What we are going to do is a bit of algebra,"},{"Start":"00:13.440 ","End":"00:17.830","Text":"some trigonometrical identities try and simplify it."},{"Start":"00:18.590 ","End":"00:22.005","Text":"What I find that works,"},{"Start":"00:22.005 ","End":"00:25.830","Text":"is if I split this up into set of"},{"Start":"00:25.830 ","End":"00:31.910","Text":"sine cubed decks to put it as sine x times sine squared x."},{"Start":"00:31.910 ","End":"00:36.440","Text":"Because then I can do a lot with sine squared x as you\u0027ll see in a minute,"},{"Start":"00:36.440 ","End":"00:39.665","Text":"1 minus cosine x, dx."},{"Start":"00:39.665 ","End":"00:44.180","Text":"Now, because we have a formula that sine"},{"Start":"00:44.180 ","End":"00:51.664","Text":"squared x plus cosine squared x equals 1,"},{"Start":"00:51.664 ","End":"00:58.030","Text":"then instead of sine squared x I could put 1 minus cosine squared x,"},{"Start":"00:58.030 ","End":"01:01.590","Text":"and you\u0027ll soon see how this helps me."},{"Start":"01:01.590 ","End":"01:04.430","Text":"Sine x, and instead of sine x,"},{"Start":"01:04.430 ","End":"01:09.620","Text":"1 minus cosine squared x,"},{"Start":"01:09.620 ","End":"01:16.240","Text":"over 1 minus cosine x, dx."},{"Start":"01:16.240 ","End":"01:19.245","Text":"Some of you will spot it right away,"},{"Start":"01:19.245 ","End":"01:24.170","Text":"but what we can do is look at this as a difference of squares formula."},{"Start":"01:24.170 ","End":"01:26.780","Text":"In other words if I write 1 as 1 squared,"},{"Start":"01:26.780 ","End":"01:27.950","Text":"you\u0027ll see it easier."},{"Start":"01:27.950 ","End":"01:36.580","Text":"Basically what we have here is a squared minus b squared over a minus b."},{"Start":"01:38.570 ","End":"01:45.055","Text":"Well, let\u0027s just do it. This is a minus b times a plus b,"},{"Start":"01:45.055 ","End":"01:47.910","Text":"over a minus b,"},{"Start":"01:47.910 ","End":"01:51.620","Text":"and then this bit cancels and we\u0027re left with a plus b."},{"Start":"01:51.620 ","End":"01:55.680","Text":"If we apply that over here,"},{"Start":"01:58.250 ","End":"02:03.455","Text":"then what we\u0027ll get is the integral"},{"Start":"02:03.455 ","End":"02:11.380","Text":"of sine x, times 1 plus cosine x,"},{"Start":"02:12.620 ","End":"02:18.105","Text":"dx, because this a minus b is canceled, this as here."},{"Start":"02:18.105 ","End":"02:22.850","Text":"Which I can then just multiply out,"},{"Start":"02:22.850 ","End":"02:26.915","Text":"expand and get the integral of sine x"},{"Start":"02:26.915 ","End":"02:35.170","Text":"plus sine x times cosine x, all this dx."},{"Start":"02:36.250 ","End":"02:42.514","Text":"Now, from here, I\u0027ll break it up into 2 bits."},{"Start":"02:42.514 ","End":"02:46.730","Text":"The sine x we know already how to do the sine x cosine x is not immediately"},{"Start":"02:46.730 ","End":"02:51.720","Text":"clear but if you remember that we have a formula that,"},{"Start":"02:55.310 ","End":"02:57.840","Text":"could be Alpha just write it as x,"},{"Start":"02:57.840 ","End":"03:06.640","Text":"2 sine x cosine x is equal to sine of 2x."},{"Start":"03:08.420 ","End":"03:13.590","Text":"If I look at sine x cosine x,"},{"Start":"03:13.590 ","End":"03:18.030","Text":"just divide by 2 basically is what I\u0027m saying sine x cosine x"},{"Start":"03:18.030 ","End":"03:22.385","Text":"is 1/2 sine of 2x if I use this here,"},{"Start":"03:22.385 ","End":"03:25.020","Text":"then I get the integral."},{"Start":"03:25.390 ","End":"03:32.170","Text":"I\u0027ll write it as a separate integrals integral of sine x, dx,"},{"Start":"03:32.170 ","End":"03:34.160","Text":"plus 1/2 and I\u0027ll put"},{"Start":"03:34.160 ","End":"03:45.890","Text":"the plus"},{"Start":"03:45.890 ","End":"03:47.044","Text":"1/2 integral"},{"Start":"03:47.044 ","End":"03:53.790","Text":"of sine 2x, dx."},{"Start":"03:53.790 ","End":"04:02.300","Text":"Now all I need to do is remind you of the integral of sine ax,"},{"Start":"04:02.300 ","End":"04:06.830","Text":"or ax plus b, is what you would expect."},{"Start":"04:06.830 ","End":"04:11.765","Text":"The integral of sine is minus cosine but here I have minus 1 over"},{"Start":"04:11.765 ","End":"04:21.000","Text":"a times cosine ax, plus constant."},{"Start":"04:21.000 ","End":"04:23.030","Text":"If I apply that here,"},{"Start":"04:23.030 ","End":"04:26.524","Text":"what we\u0027re going to get is the integral of sine,"},{"Start":"04:26.524 ","End":"04:29.525","Text":"sine on its own is minus cosine,"},{"Start":"04:29.525 ","End":"04:32.430","Text":"so minus cosine x,"},{"Start":"04:32.430 ","End":"04:34.865","Text":"and here I have plus."},{"Start":"04:34.865 ","End":"04:41.650","Text":"Now using this formula I need the minus 1 over a a here is 2,"},{"Start":"04:41.650 ","End":"04:45.929","Text":"or maybe just to highlight in the formula,"},{"Start":"04:45.929 ","End":"04:48.130","Text":"when we have an a here,"},{"Start":"04:48.130 ","End":"04:56.115","Text":"then we put an a in the denominator and here our a happens to be 2,"},{"Start":"04:56.115 ","End":"05:01.530","Text":"we get 1/2 from here,"},{"Start":"05:01.530 ","End":"05:05.380","Text":"minus 1/2 from this,"},{"Start":"05:05.750 ","End":"05:10.660","Text":"and cosine of 2x,"},{"Start":"05:12.260 ","End":"05:17.730","Text":"plus C. This is the answer except that we"},{"Start":"05:17.730 ","End":"05:23.260","Text":"might just to be a little bit tidier write it as,"},{"Start":"05:23.460 ","End":"05:26.950","Text":"minus cosine x and"},{"Start":"05:26.950 ","End":"05:36.390","Text":"then minus 1/4 cosine 2x plus C just to combine these."},{"Start":"05:36.390 ","End":"05:40.030","Text":"That\u0027s it for part A."},{"Start":"05:40.700 ","End":"05:45.025","Text":"Now I\u0027ve copied part B over here,"},{"Start":"05:45.025 ","End":"05:51.375","Text":"and I can think of some tools that could help me to modify this,"},{"Start":"05:51.375 ","End":"05:53.875","Text":"for example the numerator,"},{"Start":"05:53.875 ","End":"05:56.950","Text":"if I write it as 1 cubed,"},{"Start":"05:56.950 ","End":"06:04.475","Text":"then I have the formula from algebra that a cubed plus b cubed is equal to"},{"Start":"06:04.475 ","End":"06:09.365","Text":"a plus b times a squared"},{"Start":"06:09.365 ","End":"06:14.985","Text":"minus ab plus b squared and,"},{"Start":"06:14.985 ","End":"06:19.220","Text":"for the denominator we have a trigonometric formula that can help me,"},{"Start":"06:19.220 ","End":"06:29.565","Text":"and that is that the cosine squared of Alpha is 1/2 of 1 plus cosine 2 Alpha."},{"Start":"06:29.565 ","End":"06:33.830","Text":"Well let\u0027s see what I get if I apply these formulas."},{"Start":"06:33.830 ","End":"06:41.520","Text":"The numerator becomes, if I take a as 1 of course and b as"},{"Start":"06:41.520 ","End":"06:51.130","Text":"cosine x I get integral a plus b part is 1 plus cosine x,"},{"Start":"06:51.560 ","End":"07:01.270","Text":"times 1 squared which is 1 minus 1 times cosine x,"},{"Start":"07:02.090 ","End":"07:08.805","Text":"plus b squared which is plus cosine squared x."},{"Start":"07:08.805 ","End":"07:13.610","Text":"That\u0027s the numerator and the denominator"},{"Start":"07:13.610 ","End":"07:20.160","Text":"becomes using the trig formula I let alpha equals x over 2,"},{"Start":"07:20.160 ","End":"07:24.570","Text":"so I get 1/2 of"},{"Start":"07:24.570 ","End":"07:34.500","Text":"1 plus cosine x. I\u0027m sorry,"},{"Start":"07:34.500 ","End":"07:40.020","Text":"this is 2 Alpha, 1 second, there."},{"Start":"07:40.020 ","End":"07:44.190","Text":"That\u0027s 2 Alpha and we said that Alpha is x over 2,"},{"Start":"07:44.190 ","End":"07:49.930","Text":"this is 1 plus cosine x, dx."},{"Start":"07:49.930 ","End":"07:57.515","Text":"Already something good has happened in that 1 plus cosine x cancels with 1 plus cosine x."},{"Start":"07:57.515 ","End":"08:03.465","Text":"What I\u0027m left with is the integral."},{"Start":"08:03.465 ","End":"08:07.850","Text":"Now, dividing by 1/2 is like multiplying by 2 and I"},{"Start":"08:07.850 ","End":"08:13.285","Text":"can take that outside the brackets outside the integral."},{"Start":"08:13.285 ","End":"08:22.565","Text":"Here I have 1 minus cosine x and instead of cosine squared x,"},{"Start":"08:22.565 ","End":"08:24.935","Text":"I can use this formula again,"},{"Start":"08:24.935 ","End":"08:31.625","Text":"this time with Alpha being just x. I get plus"},{"Start":"08:31.625 ","End":"08:36.405","Text":"1/2 of 1 plus cosine"},{"Start":"08:36.405 ","End":"08:44.770","Text":"2 Alpha is 1 plus cosine this time 2x, dx."},{"Start":"08:46.800 ","End":"08:52.880","Text":"If I just open the brackets here,"},{"Start":"08:53.160 ","End":"09:00.010","Text":"I think I might even put the 2 inside 2 cancel with the 1.5,"},{"Start":"09:00.010 ","End":"09:03.640","Text":"what I\u0027ll get is the integral."},{"Start":"09:03.640 ","End":"09:07.720","Text":"I want do the arithmetic in my head,"},{"Start":"09:07.720 ","End":"09:15.325","Text":"we will get 2 times 1 from here and here we\u0027ll have 2 times 0.5, which is 1."},{"Start":"09:15.325 ","End":"09:19.075","Text":"Altogether, I\u0027ll get 3 in the number part."},{"Start":"09:19.075 ","End":"09:21.085","Text":"In the cosine x part,"},{"Start":"09:21.085 ","End":"09:28.480","Text":"I\u0027ll just get minus 2 cosine x and in the cosine 2x part,"},{"Start":"09:28.480 ","End":"09:30.865","Text":"there\u0027s 2 with this 1.5 still."},{"Start":"09:30.865 ","End":"09:36.355","Text":"It\u0027s plus cosine 2x."},{"Start":"09:36.355 ","End":"09:37.810","Text":"If I\u0027ve done my arithmetic right."},{"Start":"09:37.810 ","End":"09:40.420","Text":"Let\u0027s just see twice 1 plus 0.5."},{"Start":"09:40.420 ","End":"09:43.240","Text":"That gives me the number minus 2 cosine x."},{"Start":"09:43.240 ","End":"09:45.055","Text":"Here\u0027s 2 with 0.5."},{"Start":"09:45.055 ","End":"09:53.620","Text":"This looks fine and I\u0027ll make a bit more space here."},{"Start":"09:53.620 ","End":"09:57.445","Text":"Each of these pieces can be done separately."},{"Start":"09:57.445 ","End":"10:04.460","Text":"We get the integral of 3 is just 3x."},{"Start":"10:04.560 ","End":"10:14.905","Text":"The integral of cosine x is sine x. I have minus 2 sine x."},{"Start":"10:14.905 ","End":"10:18.055","Text":"The integral of cosine 2x."},{"Start":"10:18.055 ","End":"10:28.480","Text":"We have to use that formula which says that the integral of cosine ax/dx is equal"},{"Start":"10:28.480 ","End":"10:37.555","Text":"to 1 over A times sine ax"},{"Start":"10:37.555 ","End":"10:44.140","Text":"plus C. Here our A is equal to 2,"},{"Start":"10:44.140 ","End":"10:52.195","Text":"we just get 1 over 2 sine 2x."},{"Start":"10:52.195 ","End":"10:56.090","Text":"I just need 1 single C at the end."},{"Start":"10:56.700 ","End":"11:00.860","Text":"This is the answer to part b."},{"Start":"11:01.310 ","End":"11:05.670","Text":"Here I\u0027ve copied the Exercise C,"},{"Start":"11:05.670 ","End":"11:08.850","Text":"and I\u0027ve written a couple of formulas."},{"Start":"11:08.850 ","End":"11:11.205","Text":"They usually come in as a pair,"},{"Start":"11:11.205 ","End":"11:15.095","Text":"for sine squared Alpha and for cosine squared Alpha,"},{"Start":"11:15.095 ","End":"11:16.810","Text":"just what I\u0027ve written here."},{"Start":"11:16.810 ","End":"11:19.015","Text":"This will be very useful to us."},{"Start":"11:19.015 ","End":"11:21.970","Text":"Because here, first of all,"},{"Start":"11:21.970 ","End":"11:26.155","Text":"we have the sine squared x."},{"Start":"11:26.155 ","End":"11:28.660","Text":"If we take Alpha as x,"},{"Start":"11:28.660 ","End":"11:33.110","Text":"we get 1.5,1 minus cosine 2 x."},{"Start":"11:37.020 ","End":"11:39.205","Text":"In the other bit,"},{"Start":"11:39.205 ","End":"11:44.260","Text":"the cosine^4 is just cosine squared squared."},{"Start":"11:44.260 ","End":"11:49.570","Text":"I can take the cosine squared part with Alpha"},{"Start":"11:49.570 ","End":"11:56.560","Text":"equals x as 1.5 of 1 plus cosine 2x,"},{"Start":"11:56.560 ","End":"12:02.690","Text":"but squared, because cosine^4 and the 1/3 is cosine squared squared."},{"Start":"12:04.140 ","End":"12:14.635","Text":"What I\u0027m going to do next is see that this is 0.5 times 0.5 squared."},{"Start":"12:14.635 ","End":"12:16.780","Text":"Just put the numbers together."},{"Start":"12:16.780 ","End":"12:19.735","Text":"0.5 times 1.5 squared,"},{"Start":"12:19.735 ","End":"12:21.939","Text":"again under the integral,"},{"Start":"12:21.939 ","End":"12:28.885","Text":"times 1 minus cosine 2x,"},{"Start":"12:28.885 ","End":"12:37.760","Text":"1 plus cosine 2x squared dx."},{"Start":"12:38.010 ","End":"12:41.650","Text":"Next, I\u0027ll just multiply this out."},{"Start":"12:41.650 ","End":"12:43.330","Text":"This comes out to 1/8,"},{"Start":"12:43.330 ","End":"12:51.520","Text":"and I can put it in front of the integral sign times the integral of 1 minus cosine 2x."},{"Start":"12:51.520 ","End":"12:54.080","Text":"Here I\u0027ll just expand."},{"Start":"12:54.450 ","End":"12:58.480","Text":"If I expand this like A plus b squared,"},{"Start":"12:58.480 ","End":"13:09.590","Text":"then I get 1 squared plus twice 1 times cosine 2x plus cosine of 2x squared."},{"Start":"13:10.830 ","End":"13:13.825","Text":"There\u0027s a 2 here, it\u0027s twice."},{"Start":"13:13.825 ","End":"13:20.960","Text":"Cosine A squared 2x of this dx."},{"Start":"13:21.270 ","End":"13:24.190","Text":"This looks like it could get messy."},{"Start":"13:24.190 ","End":"13:28.490","Text":"I\u0027ll give myself fold the space I need here."},{"Start":"13:31.500 ","End":"13:37.689","Text":"I think what we\u0027ll do next is use again the cosine squared Alpha formula here."},{"Start":"13:37.689 ","End":"13:40.330","Text":"Let me try and speed things up."},{"Start":"13:40.330 ","End":"13:45.710","Text":"1/8 integral of 1 minus cosine 2x,"},{"Start":"13:45.930 ","End":"13:55.285","Text":"1 plus 2, cosine 2x plus 1.5."},{"Start":"13:55.285 ","End":"13:57.385","Text":"Now I\u0027m expanding this."},{"Start":"13:57.385 ","End":"14:01.255","Text":"1.5 of 1 plus"},{"Start":"14:01.255 ","End":"14:08.680","Text":"cosine 4x dx."},{"Start":"14:08.680 ","End":"14:14.750","Text":"Now just expanding 1/8 integral."},{"Start":"14:16.050 ","End":"14:19.450","Text":"Maybe I\u0027ll just keep this bit first,"},{"Start":"14:19.450 ","End":"14:23.335","Text":"cosine 2x and then just simplify this bracket."},{"Start":"14:23.335 ","End":"14:26.155","Text":"Where I get 1 plus 0.5,"},{"Start":"14:26.155 ","End":"14:33.820","Text":"that\u0027s 3 over 2 plus 2 cosine 2x"},{"Start":"14:33.820 ","End":"14:40.585","Text":"plus 1.5 cosine 4x."},{"Start":"14:40.585 ","End":"14:48.775","Text":"1.5 cosine 4x can\u0027t avoid it."},{"Start":"14:48.775 ","End":"14:53.455","Text":"I have to multiply out."},{"Start":"14:53.455 ","End":"14:58.930","Text":"I\u0027ll start with the 1 times all this."},{"Start":"14:58.930 ","End":"15:00.535","Text":"That\u0027s just copy this bracket."},{"Start":"15:00.535 ","End":"15:07.135","Text":"It\u0027s 3 over 2 plus 2 cosine 2x"},{"Start":"15:07.135 ","End":"15:12.130","Text":"plus 1.5 cosine 4x."},{"Start":"15:12.130 ","End":"15:18.640","Text":"Then I have to take the minus cosine 2x times each 1 of these."},{"Start":"15:18.640 ","End":"15:28.460","Text":"The minus cosine 2x is minus 3 over 2 cosine 2x."},{"Start":"15:28.740 ","End":"15:35.930","Text":"Here, we get minus 2 cosine squared 2x."},{"Start":"15:38.550 ","End":"15:42.730","Text":"It\u0027s quite clear you won\u0027t get this in an exam or anything."},{"Start":"15:42.730 ","End":"15:45.280","Text":"I don\u0027t know who thought of this diabolical exercise,"},{"Start":"15:45.280 ","End":"15:47.575","Text":"but we\u0027ll just plot on keep going."},{"Start":"15:47.575 ","End":"15:49.870","Text":"Minus 2 cosine squared 2x."},{"Start":"15:49.870 ","End":"15:54.085","Text":"The last 1 will be minus 0.5"},{"Start":"15:54.085 ","End":"16:01.620","Text":"cosine 2x cosine 4x"},{"Start":"16:01.620 ","End":"16:09.310","Text":"cosine 2x cosine 4x dx."},{"Start":"16:09.310 ","End":"16:17.780","Text":"Boy, oh boy. Now let\u0027s see if we can collect terms together."},{"Start":"16:18.570 ","End":"16:23.180","Text":"Now if I take the cosine 2x,"},{"Start":"16:25.080 ","End":"16:28.405","Text":"we have 2 minus 1.5."},{"Start":"16:28.405 ","End":"16:36.200","Text":"That\u0027s 0.5, 1.5 cosine 2x."},{"Start":"16:36.420 ","End":"16:40.820","Text":"Then there\u0027s a 0.5 cosine 4x."},{"Start":"16:45.270 ","End":"16:48.535","Text":"Now we have a couple of mixed terms,"},{"Start":"16:48.535 ","End":"16:50.860","Text":"or we have this cosine squared."},{"Start":"16:50.860 ","End":"16:54.925","Text":"Once again, we can use this formula,"},{"Start":"16:54.925 ","End":"16:58.090","Text":"the minus 2 here and there\u0027s a 1/2 here."},{"Start":"16:58.090 ","End":"17:05.890","Text":"I\u0027ll get minus 1 and minus 1 with alpha being 2x, it\u0027s minus."},{"Start":"17:06.650 ","End":"17:10.545","Text":"Minus 1 minus cosine 4x."},{"Start":"17:10.545 ","End":"17:20.590","Text":"I would have had minus 2,"},{"Start":"17:20.590 ","End":"17:27.530","Text":"but with the 1/2 it\u0027s minus 1 and minus 1 of all this brackets is minus this, minus this."},{"Start":"17:27.870 ","End":"17:35.650","Text":"Minus 1/2, and I might do this 1 at the side,"},{"Start":"17:35.650 ","End":"17:38.635","Text":"the cosine 2x, cosine 4x."},{"Start":"17:38.635 ","End":"17:45.700","Text":"Well, I\u0027ll just tell you verbally that when you have the product of cosines,"},{"Start":"17:45.700 ","End":"17:52.435","Text":"the answer is 1/2 times cosine of the sum of the angles."},{"Start":"17:52.435 ","End":"18:01.225","Text":"That\u0027s cosine of 6x plus,"},{"Start":"18:01.225 ","End":"18:04.645","Text":"wait a minute, I need brackets here."},{"Start":"18:04.645 ","End":"18:10.390","Text":"Cosine 6x plus cosine of the difference of the angles."},{"Start":"18:10.390 ","End":"18:16.490","Text":"I can take the 4x minus 2x, that\u0027s cosine 2x."},{"Start":"18:16.890 ","End":"18:25.210","Text":"Then here all this in brackets, dx."},{"Start":"18:25.210 ","End":"18:27.470","Text":"This is a nightmare."},{"Start":"18:27.630 ","End":"18:31.030","Text":"We\u0027ll just go along now, continue with it."},{"Start":"18:31.030 ","End":"18:36.530","Text":"Obviously won\u0027t get an exercise like this in the exam or anything."},{"Start":"18:36.600 ","End":"18:40.040","Text":"I\u0027m continuing anyway."},{"Start":"18:43.920 ","End":"18:47.500","Text":"Let\u0027s see what we have here now."},{"Start":"18:47.500 ","End":"18:51.460","Text":"We\u0027ll collect all the like terms together,"},{"Start":"18:51.460 ","End":"18:53.780","Text":"except that I will,"},{"Start":"18:54.600 ","End":"18:59.060","Text":"you know what, I won\u0027t even keep the 1/8."},{"Start":"18:59.160 ","End":"19:02.980","Text":"We just have 4 kinds of terms, constants,"},{"Start":"19:02.980 ","End":"19:04.645","Text":"things with cosine 2x,"},{"Start":"19:04.645 ","End":"19:07.195","Text":"with cosine 4x, and with cosine 6x."},{"Start":"19:07.195 ","End":"19:11.575","Text":"We\u0027ll just collect them all as we can."},{"Start":"19:11.575 ","End":"19:13.765","Text":"Let\u0027s take constants."},{"Start":"19:13.765 ","End":"19:22.700","Text":"Constants I have 1/8 times 3/2 minus 1,"},{"Start":"19:22.700 ","End":"19:29.360","Text":"and that\u0027s it for constant."},{"Start":"19:29.360 ","End":"19:32.500","Text":"Next we\u0027ll go with cosine 2x."},{"Start":"19:32.500 ","End":"19:38.695","Text":"Cosine 2x, I have 1/8 from here,"},{"Start":"19:38.695 ","End":"19:42.970","Text":"and then I have 1.5 because of"},{"Start":"19:42.970 ","End":"19:49.940","Text":"this cosine 2x and here minus 1/4."},{"Start":"19:50.430 ","End":"19:58.195","Text":"It\u0027s 1/2 minus a 1/4, cosine 2x."},{"Start":"19:58.195 ","End":"20:00.790","Text":"This is for the cosine 2x."},{"Start":"20:00.790 ","End":"20:02.695","Text":"Next, the cosine 4x,"},{"Start":"20:02.695 ","End":"20:06.865","Text":"cosine 4x the 1/8."},{"Start":"20:06.865 ","End":"20:10.615","Text":"Then we have plus 1/2 from here,"},{"Start":"20:10.615 ","End":"20:13.675","Text":"minus 1 from here,"},{"Start":"20:13.675 ","End":"20:18.400","Text":"and there is no cosine 4x in here."},{"Start":"20:18.400 ","End":"20:24.820","Text":"Cosine 4x. Then the cosine 6x,"},{"Start":"20:24.820 ","End":"20:32.965","Text":"I just get 1/8 minus"},{"Start":"20:32.965 ","End":"20:36.835","Text":"1/2 times 1/2"},{"Start":"20:36.835 ","End":"20:43.780","Text":"cosine 6x."},{"Start":"20:43.780 ","End":"20:53.630","Text":"It\u0027s the integral of this whole thing, dx."},{"Start":"20:55.290 ","End":"21:03.830","Text":"Next thing to do is to remember the formula that the integral,"},{"Start":"21:04.110 ","End":"21:07.355","Text":"we may do it in a different color."},{"Start":"21:07.355 ","End":"21:18.420","Text":"That the integral of cosine of ax dx"},{"Start":"21:18.420 ","End":"21:25.225","Text":"is 1/a times sine"},{"Start":"21:25.225 ","End":"21:29.190","Text":"of ax plus constant,"},{"Start":"21:29.190 ","End":"21:32.460","Text":"though we put 1 single constant at the end."},{"Start":"21:32.460 ","End":"21:36.040","Text":"Now continuing with this, we get,"},{"Start":"21:36.040 ","End":"21:40.050","Text":"first of all, the constant 3/2 minus 1 is a 1/2."},{"Start":"21:40.050 ","End":"21:43.455","Text":"1/2 times an 1/8 is 1/16."},{"Start":"21:43.455 ","End":"21:46.725","Text":"Next, the cosine 2x."},{"Start":"21:46.725 ","End":"21:54.060","Text":"A half minus 1/4 is 1/4 times 1/8 is 1/32."},{"Start":"21:54.060 ","End":"21:58.720","Text":"1/32 but I have this a here,"},{"Start":"21:58.720 ","End":"22:00.535","Text":"have to divide by a,"},{"Start":"22:00.535 ","End":"22:05.920","Text":"so it\u0027s 1/64"},{"Start":"22:05.920 ","End":"22:11.500","Text":"times sine 2x."},{"Start":"22:11.500 ","End":"22:13.135","Text":"In the next 1 with the 4x,"},{"Start":"22:13.135 ","End":"22:15.175","Text":"the a will be 4."},{"Start":"22:15.175 ","End":"22:16.915","Text":"I\u0027ll get, let\u0027s see,"},{"Start":"22:16.915 ","End":"22:20.200","Text":"1/2 minus 1 is minus 1/2,"},{"Start":"22:20.200 ","End":"22:26.860","Text":"minus 1/2 times the 1/8 is minus 1/16."},{"Start":"22:26.860 ","End":"22:30.760","Text":"Minus 1/16, but a quarter of that is"},{"Start":"22:30.760 ","End":"22:41.940","Text":"minus 1/64 sine"},{"Start":"22:41.940 ","End":"22:46.450","Text":"of 4x."},{"Start":"22:46.450 ","End":"22:49.990","Text":"Finally, the last 1 is see,"},{"Start":"22:49.990 ","End":"22:57.220","Text":"minus 1/2 times 1/2 is minus a quarter times an 1/8, minus 1/32."},{"Start":"22:57.220 ","End":"23:05.635","Text":"Minus 1/32, but I have to multiply that by 1/6."},{"Start":"23:05.635 ","End":"23:09.100","Text":"That will be minus 1 over,"},{"Start":"23:09.100 ","End":"23:17.450","Text":"I\u0027ll just write it for a minute as 32 times 6 cosine 6x."},{"Start":"23:19.470 ","End":"23:26.350","Text":"Then plus C. Let me just for a moment go to my calculator."},{"Start":"23:26.350 ","End":"23:30.265","Text":"I made it 192, which I\u0027ve put here,"},{"Start":"23:30.265 ","End":"23:32.260","Text":"and I\u0027m` done with this."},{"Start":"23:32.260 ","End":"23:34.810","Text":"There might be arithmetical errors,"},{"Start":"23:34.810 ","End":"23:41.300","Text":"but you\u0027ve got the general idea and you hope you don\u0027t get 1 of these on the exam."},{"Start":"23:42.960 ","End":"23:49.880","Text":"That concludes part c and which is the last part and we\u0027re done. Thank God."}],"ID":6723}],"Thumbnail":null,"ID":1608},{"Name":"Trigonometric Integrals Using Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Solution by Trigonometric Substitution","Duration":"13m 9s","ChapterTopicVideoID":1669,"CourseChapterTopicPlaylistID":1607,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.010","Text":"In this clip, I\u0027m going to talk about how to solve"},{"Start":"00:03.010 ","End":"00:08.425","Text":"certain trigonometric integrals by using trigonometric substitutions."},{"Start":"00:08.425 ","End":"00:09.805","Text":"Can\u0027t always be done."},{"Start":"00:09.805 ","End":"00:11.215","Text":"But when it can be done,"},{"Start":"00:11.215 ","End":"00:15.280","Text":"it\u0027s preferable over the identities method because"},{"Start":"00:15.280 ","End":"00:19.870","Text":"the identities are so cumbersome and it goes so much more quickly."},{"Start":"00:19.870 ","End":"00:21.610","Text":"I\u0027m going to show you how to recognize"},{"Start":"00:21.610 ","End":"00:27.265","Text":"certain patterns of which integrals can be done this way."},{"Start":"00:27.265 ","End":"00:32.935","Text":"The first 1, I\u0027m going to show you is the following pattern."},{"Start":"00:32.935 ","End":"00:38.874","Text":"We have the integral of cosine of x times"},{"Start":"00:38.874 ","End":"00:47.690","Text":"some mess involving sine x. I wish I mean a function of sine x dx."},{"Start":"00:47.690 ","End":"00:50.120","Text":"It turns out that if we have such a case,"},{"Start":"00:50.120 ","End":"00:54.590","Text":"such a pattern that this is solved by means of letting"},{"Start":"00:54.590 ","End":"01:01.435","Text":"substitution t equals sine x,"},{"Start":"01:01.435 ","End":"01:07.280","Text":"and I\u0027ll show you a specific example of this pattern so you see what I mean."},{"Start":"01:07.280 ","End":"01:18.410","Text":"Let\u0027s take the integral of cosine x times sine squared x"},{"Start":"01:18.410 ","End":"01:24.684","Text":"plus 1 over sine x"},{"Start":"01:24.684 ","End":"01:32.210","Text":"minus e^sine x."},{"Start":"01:32.210 ","End":"01:34.970","Text":"All this dx."},{"Start":"01:34.970 ","End":"01:38.555","Text":"I\u0027d like to reiterate what we\u0027re saying here."},{"Start":"01:38.555 ","End":"01:43.580","Text":"If you have the cosine of x like it is,"},{"Start":"01:43.580 ","End":"01:47.150","Text":"and following it all together with it,"},{"Start":"01:47.150 ","End":"01:50.900","Text":"we have some function of sine x,"},{"Start":"01:50.900 ","End":"01:55.800","Text":"some expression involving just sine x as we do here."},{"Start":"01:58.330 ","End":"02:09.500","Text":"Then the substitution t equals sine x will help us greatly to solve this problem."},{"Start":"02:09.500 ","End":"02:15.100","Text":"Let\u0027s go ahead and make that substitution and actually solve the problem."},{"Start":"02:15.100 ","End":"02:23.260","Text":"If we let t equals sine x,"},{"Start":"02:23.800 ","End":"02:28.130","Text":"what that will give us is that we also have to"},{"Start":"02:28.130 ","End":"02:33.000","Text":"take the derivative of this to put d for each of these."},{"Start":"02:33.910 ","End":"02:36.905","Text":"Derivative of this is 1,"},{"Start":"02:36.905 ","End":"02:39.065","Text":"and we multiply that by dt."},{"Start":"02:39.065 ","End":"02:41.635","Text":"That\u0027s the way it works."},{"Start":"02:41.635 ","End":"02:44.825","Text":"Here, as a function of x,"},{"Start":"02:44.825 ","End":"02:48.500","Text":"the derivative of sine x is cosine x."},{"Start":"02:48.500 ","End":"02:52.350","Text":"But here we put dx."},{"Start":"02:53.150 ","End":"02:55.895","Text":"Now, if you look at it,"},{"Start":"02:55.895 ","End":"03:00.140","Text":"this yellow bits plus the dx at the end,"},{"Start":"03:00.140 ","End":"03:03.680","Text":"cosine x dx is exactly dt."},{"Start":"03:03.680 ","End":"03:09.000","Text":"What we get from here is,"},{"Start":"03:09.000 ","End":"03:10.310","Text":"we\u0027ve got the dt at the end,"},{"Start":"03:10.310 ","End":"03:11.570","Text":"I\u0027ll write it at the end."},{"Start":"03:11.570 ","End":"03:13.355","Text":"We have sine squared,"},{"Start":"03:13.355 ","End":"03:17.510","Text":"now each sine is t, so this is just t squared."},{"Start":"03:17.510 ","End":"03:20.180","Text":"This is 1 over t,"},{"Start":"03:20.180 ","End":"03:29.910","Text":"and this is minus e^t because every sign is just t. Other dt I said."},{"Start":"03:30.770 ","End":"03:36.680","Text":"Now we have a complete different integral in terms of t without x,"},{"Start":"03:36.680 ","End":"03:38.405","Text":"and later we\u0027ll substitute back."},{"Start":"03:38.405 ","End":"03:40.205","Text":"Meanwhile, let\u0027s solve this 1."},{"Start":"03:40.205 ","End":"03:45.275","Text":"This equals, let\u0027s do it here."},{"Start":"03:45.275 ","End":"03:48.589","Text":"The integral now, t squared,"},{"Start":"03:48.589 ","End":"03:51.110","Text":"pardon me, we\u0027ll just solve it right away."},{"Start":"03:51.110 ","End":"03:53.885","Text":"We don\u0027t need to put the integral sign in anymore."},{"Start":"03:53.885 ","End":"03:59.779","Text":"What we have is integral of t squared is second."},{"Start":"03:59.779 ","End":"04:03.769","Text":"Yeah, is t cubed over 3."},{"Start":"04:03.769 ","End":"04:12.495","Text":"Integral of 1 over t is this natural log of t in bars absolute value,"},{"Start":"04:12.495 ","End":"04:18.045","Text":"and e^t is just e^t and finally"},{"Start":"04:18.045 ","End":"04:24.160","Text":"plus c. Now that\u0027s not the end because the original integral was in terms of x,"},{"Start":"04:24.160 ","End":"04:27.955","Text":"so we have to substitute back t equals sine x."},{"Start":"04:27.955 ","End":"04:38.355","Text":"What we get as our final answer is sine cubed of x over"},{"Start":"04:38.355 ","End":"04:45.634","Text":"3 plus natural log absolute value of sine x"},{"Start":"04:45.634 ","End":"04:53.530","Text":"minus e^sine x plus"},{"Start":"04:53.530 ","End":"04:58.780","Text":"c. That\u0027s it for this example."},{"Start":"04:58.780 ","End":"05:05.030","Text":"This just confirms this pattern that when you have cosine x times a function of sine x,"},{"Start":"05:05.030 ","End":"05:10.455","Text":"that\u0027s the substitution, t equals sine x. I\u0027d like to point out,"},{"Start":"05:10.455 ","End":"05:12.125","Text":"just as in the example,"},{"Start":"05:12.125 ","End":"05:14.390","Text":"that after we do this substitution,"},{"Start":"05:14.390 ","End":"05:21.915","Text":"what we get in general will be the integral of f of t dt."},{"Start":"05:21.915 ","End":"05:26.195","Text":"In other words, it will always be that the cosine of x with the dx will become dt,"},{"Start":"05:26.195 ","End":"05:31.715","Text":"and this function of sine x would just be function of t. After this substitution,"},{"Start":"05:31.715 ","End":"05:35.480","Text":"this is what we\u0027ll get and then we just have a simple integral in"},{"Start":"05:35.480 ","End":"05:41.540","Text":"t. Here it was pretty obvious,"},{"Start":"05:41.540 ","End":"05:43.570","Text":"in the example that we had,"},{"Start":"05:43.570 ","End":"05:46.405","Text":"but some of the examples are not as obvious."},{"Start":"05:46.405 ","End":"05:51.295","Text":"Let\u0027s take another example where we have the integral,"},{"Start":"05:51.295 ","End":"06:00.270","Text":"an integral of cosine cubed of x dx."},{"Start":"06:00.270 ","End":"06:05.500","Text":"I just erase this 2 here because we\u0027re still under the same paradigm. This, yeah."},{"Start":"06:05.500 ","End":"06:08.949","Text":"This 1 certainly doesn\u0027t look like this pattern,"},{"Start":"06:08.949 ","End":"06:13.825","Text":"so we might have to do a little bit of work to get it to look that way."},{"Start":"06:13.825 ","End":"06:19.420","Text":"What I suggest is doing a bit of rewriting."},{"Start":"06:19.420 ","End":"06:26.105","Text":"Intuitively, I take the cosine x out here,"},{"Start":"06:26.105 ","End":"06:32.855","Text":"and that\u0027s already that bit and what I\u0027m left with is cosine squared x dx."},{"Start":"06:32.855 ","End":"06:35.930","Text":"However, this is not a function of sine x."},{"Start":"06:35.930 ","End":"06:39.280","Text":"But then I remember the trigonometric identities,"},{"Start":"06:39.280 ","End":"06:50.270","Text":"and I remember that I can write the cosine squared x as 1 minus sine squared x dx."},{"Start":"06:50.270 ","End":"06:59.030","Text":"This comes from the formula that the sine squared x plus cosine squared x equals 1."},{"Start":"06:59.030 ","End":"07:02.450","Text":"Well, if you just have cosine squared x and you"},{"Start":"07:02.450 ","End":"07:06.590","Text":"can as 1 minus sine squared by bringing that to the other side."},{"Start":"07:06.590 ","End":"07:13.200","Text":"Now this does fall under the pattern that we have."},{"Start":"07:15.190 ","End":"07:17.990","Text":"Second, yeah, we have,"},{"Start":"07:17.990 ","End":"07:21.790","Text":"this is the function of sine x,"},{"Start":"07:21.790 ","End":"07:27.650","Text":"and we have that this is the cosine of x as up here."},{"Start":"07:27.650 ","End":"07:31.490","Text":"Now it fits the pattern that if we let"},{"Start":"07:31.490 ","End":"07:39.335","Text":"t equals sine x,"},{"Start":"07:39.335 ","End":"07:47.470","Text":"then what it says here guarantees that we\u0027re going to get the integral of f of t,"},{"Start":"07:47.470 ","End":"07:53.470","Text":"which is this thing but t instead of sine x of 1 minus t squared,"},{"Start":"07:53.470 ","End":"07:54.610","Text":"every time I see sine x,"},{"Start":"07:54.610 ","End":"07:57.770","Text":"I put t dt."},{"Start":"07:58.070 ","End":"08:01.495","Text":"Now this is equal to,"},{"Start":"08:01.495 ","End":"08:06.340","Text":"this is just immediate integral of 1 is just t, I\u0027m sorry,"},{"Start":"08:06.340 ","End":"08:11.545","Text":"I put x here instead of the t and t squared"},{"Start":"08:11.545 ","End":"08:18.030","Text":"is t^3 over 3 plus constant."},{"Start":"08:18.030 ","End":"08:24.730","Text":"The only thing that remains to be done now is to instead of t put back sine of x,"},{"Start":"08:24.730 ","End":"08:32.695","Text":"so our answer will be sine of x minus"},{"Start":"08:32.695 ","End":"08:37.630","Text":"sine cubed x over"},{"Start":"08:37.630 ","End":"08:43.700","Text":"3 plus c. This is 1 famous pattern,"},{"Start":"08:43.700 ","End":"08:45.850","Text":"and just before they give the other 1, I want to make space."},{"Start":"08:45.850 ","End":"08:48.140","Text":"I\u0027ll put this over here."},{"Start":"08:48.140 ","End":"08:51.790","Text":"The second pattern is very similar,"},{"Start":"08:51.790 ","End":"08:53.905","Text":"except that instead of the cosine,"},{"Start":"08:53.905 ","End":"08:56.360","Text":"we\u0027ll have the sign."},{"Start":"08:57.090 ","End":"09:06.220","Text":"We\u0027ll have some function of cosine x dx."},{"Start":"09:06.220 ","End":"09:12.790","Text":"This time the substitution we\u0027ll make will be t equals cosine x,"},{"Start":"09:12.790 ","End":"09:19.490","Text":"and what this will give us is that we"},{"Start":"09:19.490 ","End":"09:26.405","Text":"get the integral of minus f of t dt."},{"Start":"09:26.405 ","End":"09:28.865","Text":"Why the minus here?"},{"Start":"09:28.865 ","End":"09:33.055","Text":"Because if we make t equals sine x,"},{"Start":"09:33.055 ","End":"09:35.960","Text":"we get dt is cosine x."},{"Start":"09:35.960 ","End":"09:37.625","Text":"But when we differentiate cosine,"},{"Start":"09:37.625 ","End":"09:39.649","Text":"we get minus sign,"},{"Start":"09:39.649 ","End":"09:42.980","Text":"so we need the extra minus here."},{"Start":"09:42.980 ","End":"09:47.450","Text":"Let\u0027s do a specific example with this pattern."},{"Start":"09:48.090 ","End":"09:58.460","Text":"Let\u0027s take the integral of sine x times cosine^4"},{"Start":"09:58.460 ","End":"10:08.950","Text":"x minus 1 over cosine x plus 4 cosine x."},{"Start":"10:10.370 ","End":"10:19.630","Text":"We\u0027ll let t equals cosine x,"},{"Start":"10:20.630 ","End":"10:23.495","Text":"and according to what\u0027s written here,"},{"Start":"10:23.495 ","End":"10:29.645","Text":"we now have to evaluate the integral of minus."},{"Start":"10:29.645 ","End":"10:37.055","Text":"Well, the minus I can put in front of f of t,"},{"Start":"10:37.055 ","End":"10:42.590","Text":"which is t^4 minus 1 over"},{"Start":"10:42.590 ","End":"10:50.780","Text":"t plus 4t dt."},{"Start":"10:50.780 ","End":"10:52.880","Text":"This is pretty immediate."},{"Start":"10:52.880 ","End":"11:02.120","Text":"What we get is minus t^4 is cosine^4."},{"Start":"11:04.380 ","End":"11:08.930","Text":"I\u0027m sorry. It\u0027s t^5 over 5."},{"Start":"11:10.060 ","End":"11:16.490","Text":"1 over t gives us natural log of absolute value of"},{"Start":"11:16.490 ","End":"11:23.305","Text":"t. 4t gives us,"},{"Start":"11:23.305 ","End":"11:25.090","Text":"we raise the power by 1,"},{"Start":"11:25.090 ","End":"11:27.100","Text":"it\u0027s t squared and divide by that,"},{"Start":"11:27.100 ","End":"11:35.674","Text":"so we get 2t squared plus a constant."},{"Start":"11:35.674 ","End":"11:40.630","Text":"At the very end, we now substitute back and instead of t,"},{"Start":"11:40.630 ","End":"11:42.970","Text":"we put cosine x."},{"Start":"11:42.970 ","End":"11:46.520","Text":"We have minus cosine^5x"},{"Start":"11:47.250 ","End":"11:55.254","Text":"over 5 plus natural log"},{"Start":"11:55.254 ","End":"12:02.355","Text":"of absolute value of cosine x."},{"Start":"12:02.355 ","End":"12:04.970","Text":"Because it\u0027s a minus and a minus is a plus,"},{"Start":"12:04.970 ","End":"12:07.590","Text":"and here we have again a minus."},{"Start":"12:08.920 ","End":"12:14.225","Text":"We have minus 2 and t squared is"},{"Start":"12:14.225 ","End":"12:21.340","Text":"cosine squared of x and plus the constant."},{"Start":"12:21.680 ","End":"12:29.190","Text":"That\u0027s it for number 2, for the example."},{"Start":"12:29.190 ","End":"12:33.725","Text":"We\u0027ve talked about 2 basic patterns."},{"Start":"12:33.725 ","End":"12:36.290","Text":"Cosine x times a function of sine x,"},{"Start":"12:36.290 ","End":"12:40.550","Text":"and we just now talked about sine x times a function of cosine x."},{"Start":"12:40.550 ","End":"12:42.020","Text":"We gave a couple of examples here."},{"Start":"12:42.020 ","End":"12:44.165","Text":"We just did an example here."},{"Start":"12:44.165 ","End":"12:47.135","Text":"There are many more patterns,"},{"Start":"12:47.135 ","End":"12:51.170","Text":"but I\u0027m splitting this into part A and part B."},{"Start":"12:51.170 ","End":"12:55.350","Text":"This is just part A,"},{"Start":"12:55.850 ","End":"13:01.845","Text":"and others will be continued in part B."},{"Start":"13:01.845 ","End":"13:04.380","Text":"That this is it for now."},{"Start":"13:04.380 ","End":"13:07.570","Text":"Coming soon part B."}],"ID":1677},{"Watched":false,"Name":"Exercise 1","Duration":"6m 57s","ChapterTopicVideoID":6663,"CourseChapterTopicPlaylistID":1607,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to compute the following integrals, 3 of them."},{"Start":"00:04.590 ","End":"00:07.320","Text":"I\u0027ve already copied the first 1 here."},{"Start":"00:07.320 ","End":"00:10.140","Text":"After looking at it, I noticed that it fits"},{"Start":"00:10.140 ","End":"00:14.160","Text":"a certain template that we studied in the theory section."},{"Start":"00:14.160 ","End":"00:18.135","Text":"That if you have an expression involving just sine x,"},{"Start":"00:18.135 ","End":"00:22.860","Text":"and I\u0027ll take this like a function of sine x only,"},{"Start":"00:22.860 ","End":"00:27.105","Text":"multiplied by just cosine x on its own,"},{"Start":"00:27.105 ","End":"00:29.955","Text":"and there\u0027s a special substitution that will work,"},{"Start":"00:29.955 ","End":"00:35.670","Text":"and that substitution is t equals sine x."},{"Start":"00:35.670 ","End":"00:37.349","Text":"As with every substitution,"},{"Start":"00:37.349 ","End":"00:39.210","Text":"we want to differentiate it."},{"Start":"00:39.210 ","End":"00:44.810","Text":"On the left we have t so we differentiate it, we get 1."},{"Start":"00:44.810 ","End":"00:50.835","Text":"But then we multiply it by dt because this is the t side of the equation."},{"Start":"00:50.835 ","End":"00:52.670","Text":"On the other side,"},{"Start":"00:52.670 ","End":"00:55.370","Text":"we differentiate sine x,"},{"Start":"00:55.370 ","End":"00:57.110","Text":"we get cosine x."},{"Start":"00:57.110 ","End":"00:59.585","Text":"But we also, and this is the rule,"},{"Start":"00:59.585 ","End":"01:02.719","Text":"add a dx here multiplied."},{"Start":"01:02.719 ","End":"01:05.915","Text":"Now we can see what we have here,"},{"Start":"01:05.915 ","End":"01:10.910","Text":"sine x is t. What I get is,"},{"Start":"01:10.910 ","End":"01:12.350","Text":"everywhere I see sine x,"},{"Start":"01:12.350 ","End":"01:20.430","Text":"I put t so this expression in green becomes t squared plus t plus 2."},{"Start":"01:20.430 ","End":"01:24.630","Text":"The cosine of x dx is just dt."},{"Start":"01:24.630 ","End":"01:34.060","Text":"Now, I have a straightforward integral in just t. What I get is t cubed over 3,"},{"Start":"01:34.060 ","End":"01:38.025","Text":"plus t squared over 2,"},{"Start":"01:38.025 ","End":"01:41.750","Text":"plus 2t plus a constant."},{"Start":"01:41.750 ","End":"01:44.620","Text":"However, we can\u0027t stay in the world of t,"},{"Start":"01:44.620 ","End":"01:46.895","Text":"we have to get back to the world of x."},{"Start":"01:46.895 ","End":"01:49.040","Text":"We now substitute back,"},{"Start":"01:49.040 ","End":"01:51.110","Text":"t is sine x."},{"Start":"01:51.110 ","End":"01:56.445","Text":"T cubed will be sine cubed of x over 3,"},{"Start":"01:56.445 ","End":"02:02.040","Text":"plus sine squared x over 2,"},{"Start":"02:02.040 ","End":"02:05.790","Text":"plus 2 sine x from here,"},{"Start":"02:05.790 ","End":"02:08.595","Text":"and plus the constant."},{"Start":"02:08.595 ","End":"02:14.510","Text":"That\u0027s it for part A. I\u0027ve copied part B from here over here,"},{"Start":"02:14.510 ","End":"02:21.125","Text":"and I\u0027ve already highlighted it because I\u0027m going to use the same idea as in part A."},{"Start":"02:21.125 ","End":"02:26.360","Text":"Just like we have a formula when we have a function of x times cosine x,"},{"Start":"02:26.360 ","End":"02:31.925","Text":"there\u0027s an almost identical 1 for when we have a function of cosine x times sine x."},{"Start":"02:31.925 ","End":"02:35.990","Text":"Just that this time instead of substituting t equals sine x,"},{"Start":"02:35.990 ","End":"02:40.640","Text":"we substitute t equals cosine x."},{"Start":"02:40.640 ","End":"02:45.455","Text":"Then if we differentiate dt is equal to,"},{"Start":"02:45.455 ","End":"02:47.570","Text":"but this time there\u0027s a minus,"},{"Start":"02:47.570 ","End":"02:50.735","Text":"because the derivative is minus sine x,"},{"Start":"02:50.735 ","End":"02:53.140","Text":"and here, we have dx."},{"Start":"02:53.140 ","End":"02:56.180","Text":"When we substitute wherever we see cosine x,"},{"Start":"02:56.180 ","End":"02:58.490","Text":"we put t. But at the end,"},{"Start":"02:58.490 ","End":"03:00.200","Text":"when we have the sine x dx,"},{"Start":"03:00.200 ","End":"03:02.760","Text":"we have to put it as minus dt."},{"Start":"03:03.310 ","End":"03:12.705","Text":"We get the integral of t cubed plus t minus 2,"},{"Start":"03:12.705 ","End":"03:15.135","Text":"that\u0027s this green part."},{"Start":"03:15.135 ","End":"03:18.390","Text":"Now, the sine x dx is minus dt."},{"Start":"03:18.390 ","End":"03:20.775","Text":"I\u0027ll just put the dt here,"},{"Start":"03:20.775 ","End":"03:23.415","Text":"and the minus in front."},{"Start":"03:23.415 ","End":"03:34.145","Text":"Rewriting it, I\u0027ll just put it as integral of minus t cubed minus t plus 2 dt,"},{"Start":"03:34.145 ","End":"03:39.925","Text":"which now equals minus t to the fourth over 4,"},{"Start":"03:39.925 ","End":"03:44.145","Text":"minus t squared over 2,"},{"Start":"03:44.145 ","End":"03:51.000","Text":"plus 2 t plus C. Now we have to substitute back,"},{"Start":"03:51.000 ","End":"03:54.570","Text":"instead of the t we have to put back cosine x."},{"Start":"03:54.570 ","End":"04:02.435","Text":"We get minus cosine to the fourth x over 4 from here,"},{"Start":"04:02.435 ","End":"04:07.015","Text":"minus cosine squared x over"},{"Start":"04:07.015 ","End":"04:16.005","Text":"2 plus 2 cosine x plus C. We\u0027re done,"},{"Start":"04:16.005 ","End":"04:18.105","Text":"and so on to the next."},{"Start":"04:18.105 ","End":"04:21.270","Text":"I\u0027ve copied part C over here."},{"Start":"04:21.270 ","End":"04:24.660","Text":"It\u0027s possible to do this integral with"},{"Start":"04:24.660 ","End":"04:30.325","Text":"just trigonometrical identities but there is another way also with substitutions."},{"Start":"04:30.325 ","End":"04:36.530","Text":"I had somehow liked to use the same trick of formula as in part A,"},{"Start":"04:36.530 ","End":"04:38.815","Text":"but I don\u0027t have any sine x\u0027s here."},{"Start":"04:38.815 ","End":"04:41.440","Text":"I do have a cosine x dx though,"},{"Start":"04:41.440 ","End":"04:47.840","Text":"and that gives me the idea to rewrite this as the integral of"},{"Start":"04:47.840 ","End":"04:54.665","Text":"cosine squared x times cosine x dx."},{"Start":"04:54.665 ","End":"05:00.500","Text":"The idea I have is to somehow convert this bit into something containing only sine x."},{"Start":"05:00.500 ","End":"05:04.985","Text":"What comes to my rescue is the formula."},{"Start":"05:04.985 ","End":"05:14.540","Text":"I happen to know that cosine squared x equals 1 minus sine squared x."},{"Start":"05:14.540 ","End":"05:16.520","Text":"You have seen this before but if not,"},{"Start":"05:16.520 ","End":"05:20.030","Text":"then it just follows from the standard formula,"},{"Start":"05:20.030 ","End":"05:27.190","Text":"that sine squared x plus cosine squared x is equal to 1 in general."},{"Start":"05:27.190 ","End":"05:29.660","Text":"If we have cosine squared x,"},{"Start":"05:29.660 ","End":"05:32.630","Text":"then we can say it\u0027s equal to 1 minus sine squared x,"},{"Start":"05:32.630 ","End":"05:36.945","Text":"by just putting the sine squared x on the right-hand side."},{"Start":"05:36.945 ","End":"05:38.485","Text":"What we get here,"},{"Start":"05:38.485 ","End":"05:45.195","Text":"is actually the integral of 1 minus sine squared x,"},{"Start":"05:45.195 ","End":"05:49.635","Text":"times cosine x dx."},{"Start":"05:49.635 ","End":"05:52.035","Text":"Now, it is in this form,"},{"Start":"05:52.035 ","End":"05:54.960","Text":"a function of x times cosine x."},{"Start":"05:54.960 ","End":"05:57.590","Text":"We do the same substitution as before,"},{"Start":"05:57.590 ","End":"06:03.405","Text":"we say t is equal to sine x and as before,"},{"Start":"06:03.405 ","End":"06:08.790","Text":"dt equals cosine x dx."},{"Start":"06:08.790 ","End":"06:11.740","Text":"After we substitute these 2 here,"},{"Start":"06:11.740 ","End":"06:18.060","Text":"we get the integral of 1 minus t squared,"},{"Start":"06:18.060 ","End":"06:19.665","Text":"cosine x is t,"},{"Start":"06:19.665 ","End":"06:22.650","Text":"and the cosine x dx is dt."},{"Start":"06:22.650 ","End":"06:26.225","Text":"This is a straightforward integral."},{"Start":"06:26.225 ","End":"06:34.785","Text":"This equals t minus t cubed over 3 plus constant."},{"Start":"06:34.785 ","End":"06:38.540","Text":"All we have to do is convert back from t to x,"},{"Start":"06:38.540 ","End":"06:40.325","Text":"by means of this again."},{"Start":"06:40.325 ","End":"06:43.785","Text":"We have sine x minus"},{"Start":"06:43.785 ","End":"06:52.485","Text":"sine cubed x over 3 plus a constant, and that\u0027s it."},{"Start":"06:52.485 ","End":"06:57.640","Text":"That was part C, which was the last part in the set, and we\u0027re done."}],"ID":6724},{"Watched":false,"Name":"Exercise 2","Duration":"11m 27s","ChapterTopicVideoID":6664,"CourseChapterTopicPlaylistID":1607,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this exercise, we have to compute the following integrals, 3 of them."},{"Start":"00:05.010 ","End":"00:06.720","Text":"Let\u0027s go to the first 1,"},{"Start":"00:06.720 ","End":"00:12.280","Text":"which is the integral of sine cubed of 2xdx."},{"Start":"00:12.560 ","End":"00:17.160","Text":"It\u0027s possible to do it entirely with trigonometrical identities,"},{"Start":"00:17.160 ","End":"00:19.755","Text":"but I would like to use substitution here."},{"Start":"00:19.755 ","End":"00:29.445","Text":"I want to rewrite this as the integral of sine squared of 2x times sine 2x."},{"Start":"00:29.445 ","End":"00:35.570","Text":"What I have in mind is doing some special formula trick like we did with when you have"},{"Start":"00:35.570 ","End":"00:41.780","Text":"a function of sine x times a cosine x or function of cosine x times a sine x."},{"Start":"00:41.780 ","End":"00:45.725","Text":"What I want to do is convert this sine squared to just cosines."},{"Start":"00:45.725 ","End":"00:48.890","Text":"How I claim that sine squared 2x is"},{"Start":"00:48.890 ","End":"00:58.665","Text":"1 minus cosine squared 2x."},{"Start":"00:58.665 ","End":"01:01.070","Text":"I\u0027ll show you at the side why."},{"Start":"01:01.070 ","End":"01:03.530","Text":"This is because in general,"},{"Start":"01:03.530 ","End":"01:09.210","Text":"sine squared Alpha plus cosine squared Alpha, equals 1."},{"Start":"01:09.210 ","End":"01:13.530","Text":"Automatically, the sine squared Alpha is just"},{"Start":"01:13.530 ","End":"01:18.210","Text":"rewriting it as 1 minus cosine squared Alpha."},{"Start":"01:18.210 ","End":"01:21.235","Text":"Here, if I take my Alpha to be 2x,"},{"Start":"01:21.235 ","End":"01:23.510","Text":"that\u0027s exactly what I have here."},{"Start":"01:23.510 ","End":"01:27.520","Text":"Now, I also have the sine 2xdx."},{"Start":"01:27.850 ","End":"01:30.200","Text":"Now, what I have here,"},{"Start":"01:30.200 ","End":"01:33.350","Text":"if I color it, is very much like what we had before."},{"Start":"01:33.350 ","End":"01:36.770","Text":"We had a function of cosine x times sine x."},{"Start":"01:36.770 ","End":"01:39.320","Text":"This also works even if it\u0027s not x,"},{"Start":"01:39.320 ","End":"01:43.415","Text":"if it\u0027s 2x as long as it\u0027s the same here and here."},{"Start":"01:43.415 ","End":"01:48.350","Text":"I mean, I couldn\u0027t have 2x here and 3x here or something like that."},{"Start":"01:48.350 ","End":"01:49.985","Text":"As long as it\u0027s the same angle,"},{"Start":"01:49.985 ","End":"01:51.695","Text":"then we can use the same trick,"},{"Start":"01:51.695 ","End":"01:57.555","Text":"which is to substitute t equals cosine of 2x."},{"Start":"01:57.555 ","End":"02:00.000","Text":"Dt, when we differentiate,"},{"Start":"02:00.000 ","End":"02:07.050","Text":"we get 1 times dt and the derivative of cosine 2x is minus 2 sine 2x."},{"Start":"02:07.050 ","End":"02:13.680","Text":"Here, we have a dx, which means that sine 2xdx,"},{"Start":"02:13.680 ","End":"02:18.030","Text":"when I go over there will be dt over minus 2."},{"Start":"02:18.030 ","End":"02:20.940","Text":"But let\u0027s first substitute the green part."},{"Start":"02:20.940 ","End":"02:24.105","Text":"Here, we have 1 minus,"},{"Start":"02:24.105 ","End":"02:26.010","Text":"the cosine 2x is t,"},{"Start":"02:26.010 ","End":"02:31.970","Text":"so 1 minus t squared times over here, like I said,"},{"Start":"02:31.970 ","End":"02:36.440","Text":"sine 2xdx is dt over minus 2 or I could just write it as,"},{"Start":"02:36.440 ","End":"02:41.130","Text":"say, minus half times dt."},{"Start":"02:42.290 ","End":"02:46.685","Text":"This is a straightforward integral in t,"},{"Start":"02:46.685 ","End":"02:51.065","Text":"so what we get is I\u0027ll put the minus half in front,"},{"Start":"02:51.065 ","End":"02:54.595","Text":"and here I have the integral of 1 is t,"},{"Start":"02:54.595 ","End":"03:01.455","Text":"the integral of t squared is t cubed over 3, plus a constant."},{"Start":"03:01.455 ","End":"03:07.020","Text":"I could put the minus half and minus half t"},{"Start":"03:07.020 ","End":"03:15.180","Text":"plus over 6 or 1/6th of t cubed."},{"Start":"03:15.180 ","End":"03:18.080","Text":"That\u0027s the answer in the world of t,"},{"Start":"03:18.080 ","End":"03:20.495","Text":"but we have to get back to the world of x."},{"Start":"03:20.495 ","End":"03:22.945","Text":"We put t equals cosine 2x,"},{"Start":"03:22.945 ","End":"03:28.560","Text":"we get minus half cosine 2x plus"},{"Start":"03:28.560 ","End":"03:35.115","Text":"one-sixth cosine cubed 2x, plus the constant."},{"Start":"03:35.115 ","End":"03:37.395","Text":"That\u0027s it for part a."},{"Start":"03:37.395 ","End":"03:42.320","Text":"Here is part b and I want to use the same trick"},{"Start":"03:42.320 ","End":"03:47.815","Text":"again to have a function of sine x times the cosine x or vice versa."},{"Start":"03:47.815 ","End":"03:52.205","Text":"Now, I see that I have an odd number of cosines,"},{"Start":"03:52.205 ","End":"03:55.385","Text":"which makes me think I want to take 1 of those cosines"},{"Start":"03:55.385 ","End":"03:58.970","Text":"next to the dx and try and put everything else in terms of sine."},{"Start":"03:58.970 ","End":"04:00.635","Text":"My idea is this."},{"Start":"04:00.635 ","End":"04:02.335","Text":"If I take the integral,"},{"Start":"04:02.335 ","End":"04:03.960","Text":"if I\u0027m looking for sines,"},{"Start":"04:03.960 ","End":"04:06.075","Text":"already here I have sine x,"},{"Start":"04:06.075 ","End":"04:09.780","Text":"what I want to do is take cosine to the 4th x,"},{"Start":"04:09.780 ","End":"04:15.860","Text":"but I\u0027m going to write the cosine to the 4th as cosine squared squared."},{"Start":"04:15.860 ","End":"04:19.050","Text":"That will be cosine to the 4th and now that\u0027s only 4 out of 5,"},{"Start":"04:19.050 ","End":"04:22.630","Text":"so I still need the cosine xdx."},{"Start":"04:23.770 ","End":"04:26.150","Text":"The idea is just like before,"},{"Start":"04:26.150 ","End":"04:30.565","Text":"cosine squared can be written in terms of sine and vice versa."},{"Start":"04:30.565 ","End":"04:32.735","Text":"I\u0027ve kept this formula up,"},{"Start":"04:32.735 ","End":"04:37.344","Text":"which shows me that this time I can say that cosine squared"},{"Start":"04:37.344 ","End":"04:43.515","Text":"Alpha is 1 minus sine squared Alpha with Alpha being just x."},{"Start":"04:43.515 ","End":"04:50.450","Text":"What I get is that this is equal to the integral of sine to the 4thx,"},{"Start":"04:50.450 ","End":"04:56.435","Text":"1 minus sine squared x squared,"},{"Start":"04:56.435 ","End":"05:00.620","Text":"and then cosine xdx."},{"Start":"05:00.620 ","End":"05:04.520","Text":"Now as before, when you have a function of just sine x,"},{"Start":"05:04.520 ","End":"05:08.075","Text":"that\u0027s what\u0027s in the green here, times cosine x,"},{"Start":"05:08.075 ","End":"05:12.770","Text":"we make the substitution that t is equal to"},{"Start":"05:12.770 ","End":"05:20.350","Text":"sine x and then dt is equal to cosine xdx."},{"Start":"05:20.510 ","End":"05:24.485","Text":"That works very nicely because this part,"},{"Start":"05:24.485 ","End":"05:31.130","Text":"the cosine xdx is here and here we just have sine x involved in this function."},{"Start":"05:31.130 ","End":"05:34.495","Text":"When we substitute, we get the integral."},{"Start":"05:34.495 ","End":"05:40.950","Text":"In the green part we get stuff with t. We have t to the 4th times 1,"},{"Start":"05:40.950 ","End":"05:45.630","Text":"minus t squared squared,"},{"Start":"05:45.630 ","End":"05:49.510","Text":"and cosine xdx is just dt."},{"Start":"05:50.120 ","End":"05:57.080","Text":"This is an integral polynomial in t. Now we need to do a bit of algebra."},{"Start":"05:57.080 ","End":"06:03.380","Text":"First of all, I\u0027ll open the brackets for this like a minus b all"},{"Start":"06:03.380 ","End":"06:09.679","Text":"squared is a squared minus 2ab plus b squared."},{"Start":"06:09.679 ","End":"06:15.590","Text":"Using that formula, we get to this point and now I\u0027ll expand by multiplying"},{"Start":"06:15.590 ","End":"06:21.650","Text":"t to the 4th using the distributive law in arithmetic algebra."},{"Start":"06:21.650 ","End":"06:28.785","Text":"We get the integral of t to the 4th minus 2t to the 6th,"},{"Start":"06:28.785 ","End":"06:30.995","Text":"I\u0027m also using rules of exponents of course,"},{"Start":"06:30.995 ","End":"06:35.960","Text":"plus t to the 8th dt."},{"Start":"06:35.960 ","End":"06:39.555","Text":"Next, I can integrate this,"},{"Start":"06:39.555 ","End":"06:43.230","Text":"so we get t to the 5th over"},{"Start":"06:43.230 ","End":"06:51.330","Text":"5 minus 2t to the 7th over 7,"},{"Start":"06:51.330 ","End":"06:59.850","Text":"and plus t to the 9th over 9 plus constant."},{"Start":"06:59.850 ","End":"07:06.690","Text":"Finally, last step is to convert back from t to x using this."},{"Start":"07:06.690 ","End":"07:12.720","Text":"We get sine to the 5th x over"},{"Start":"07:12.720 ","End":"07:21.600","Text":"5 minus 2 sine to the 7th x over 7,"},{"Start":"07:21.600 ","End":"07:29.235","Text":"plus sine to the 9th x over 9 plus a constant."},{"Start":"07:29.235 ","End":"07:34.620","Text":"That does part b. On to the next."},{"Start":"07:34.620 ","End":"07:39.470","Text":"Here\u0027s part c. It\u0027s actually very similar to part b,"},{"Start":"07:39.470 ","End":"07:43.280","Text":"more or less the same but with sine and cosine interchanged."},{"Start":"07:43.280 ","End":"07:47.870","Text":"I\u0027m going to use the same trick if you like as before,"},{"Start":"07:47.870 ","End":"07:49.610","Text":"and write this as the integral."},{"Start":"07:49.610 ","End":"07:57.225","Text":"Now, I want to get all stuff in terms of cosine x and finally a sine x at the end."},{"Start":"07:57.225 ","End":"08:01.950","Text":"I\u0027ll take the sine x off here and it\u0027ll be sine to the 4th."},{"Start":"08:01.950 ","End":"08:12.585","Text":"Sine to the 4th is sine squared squared and sine squared is 1 minus cosine squared Alpha."},{"Start":"08:12.585 ","End":"08:20.490","Text":"Basically, what I\u0027m saying is that sine to the 5th of x is sine to"},{"Start":"08:20.490 ","End":"08:30.540","Text":"the 4th x times sine x and this sine to the 4th is sine squared squared."},{"Start":"08:30.540 ","End":"08:36.305","Text":"I have sine squared x squared"},{"Start":"08:36.305 ","End":"08:43.375","Text":"times cosine to the 4th and at the end, sine xdx."},{"Start":"08:43.375 ","End":"08:47.150","Text":"Very well. Now, I replace the sine"},{"Start":"08:47.150 ","End":"08:51.440","Text":"squared with very similar formula that we used last time."},{"Start":"08:51.440 ","End":"08:57.005","Text":"Basically, all comes from this identity that sine squared is 1 minus cosine squared."},{"Start":"08:57.005 ","End":"09:05.630","Text":"We get the integral of 1 minus cosine squared x squared,"},{"Start":"09:05.630 ","End":"09:10.459","Text":"times cosine to the 4th x times"},{"Start":"09:10.459 ","End":"09:16.050","Text":"sine xdx."},{"Start":"09:16.050 ","End":"09:21.285","Text":"This time we have a function of cosine x times sine x,"},{"Start":"09:21.285 ","End":"09:29.165","Text":"so the substitution we need is t equals cosine x and therefore,"},{"Start":"09:29.165 ","End":"09:31.745","Text":"dt, we\u0027ve seen this before,"},{"Start":"09:31.745 ","End":"09:38.300","Text":"is minus sine xdx."},{"Start":"09:38.300 ","End":"09:41.930","Text":"When I substitute, what I get is for the green part,"},{"Start":"09:41.930 ","End":"09:50.340","Text":"it\u0027s all in terms of t. We have the integral of 1 minus t squared squared,"},{"Start":"09:50.340 ","End":"09:54.675","Text":"times t to the 4th times sine xdx."},{"Start":"09:54.675 ","End":"09:58.290","Text":"It\u0027s not exactly dt, it\u0027s minus dt."},{"Start":"09:58.290 ","End":"10:06.340","Text":"I\u0027ll just put a minus here and then we have the dt here."},{"Start":"10:06.340 ","End":"10:09.520","Text":"As before, we need to do a bit of algebra."},{"Start":"10:09.520 ","End":"10:15.165","Text":"This is equal to 1 minus t squared squared is"},{"Start":"10:15.165 ","End":"10:22.685","Text":"1 minus 2t squared plus t to the 4th, times t to the 4th dt."},{"Start":"10:22.685 ","End":"10:32.045","Text":"Multiplying again, we get the integral of t to the 4th minus 2t to the 6th,"},{"Start":"10:32.045 ","End":"10:34.145","Text":"plus t to the 8th dt."},{"Start":"10:34.145 ","End":"10:41.100","Text":"The integral of this is t to the 5th over"},{"Start":"10:41.100 ","End":"10:48.870","Text":"5 minus 2t to the 7th over 7,"},{"Start":"10:48.870 ","End":"10:56.445","Text":"plus t to the 9th over 9 plus a constant."},{"Start":"10:56.445 ","End":"11:02.855","Text":"Finally, we need to convert back from t to x using this."},{"Start":"11:02.855 ","End":"11:08.070","Text":"We get cosine to the 5th x over"},{"Start":"11:08.070 ","End":"11:14.790","Text":"5 minus 2 cosine to the 7th x over 7,"},{"Start":"11:14.790 ","End":"11:22.500","Text":"plus cosine to the 9th x over 9, plus the constant."},{"Start":"11:22.500 ","End":"11:28.300","Text":"That concludes part c. We\u0027re done with this exercise."}],"ID":6725},{"Watched":false,"Name":"Exercise 3","Duration":"11m 29s","ChapterTopicVideoID":6665,"CourseChapterTopicPlaylistID":1607,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.145","Text":"In this exercise, we have to compute the following 3 integrals."},{"Start":"00:04.145 ","End":"00:08.065","Text":"Start with the first cosine to the 5th x."},{"Start":"00:08.065 ","End":"00:13.650","Text":"The idea is to use the tricks we\u0027ve used before which is to get this to be"},{"Start":"00:13.650 ","End":"00:16.440","Text":"either a function of cosine x times"},{"Start":"00:16.440 ","End":"00:20.535","Text":"sine x or the reverse of a function of sine x times cosine x."},{"Start":"00:20.535 ","End":"00:22.875","Text":"Since 5 is an odd number,"},{"Start":"00:22.875 ","End":"00:26.640","Text":"I can easily pull off a cosine x and then I\u0027ll get cosine to"},{"Start":"00:26.640 ","End":"00:31.215","Text":"the 4th and then somehow I think we can get that in terms of sine x."},{"Start":"00:31.215 ","End":"00:36.420","Text":"Before I do that, I\u0027ll also be writing a formula that I often use and that is"},{"Start":"00:36.420 ","End":"00:42.780","Text":"the cosine squared Alpha plus sine squared Alpha equals 1."},{"Start":"00:42.780 ","End":"00:46.720","Text":"For example, if I had cosine squared Alpha,"},{"Start":"00:46.720 ","End":"00:53.355","Text":"I could put it in terms of sine Alpha as 1 minus sine squared Alpha."},{"Start":"00:53.355 ","End":"00:59.275","Text":"Here, cosine to the 5th could be cosine to the 4th times cosine."},{"Start":"00:59.275 ","End":"01:01.120","Text":"I\u0027ll even do another step."},{"Start":"01:01.120 ","End":"01:05.760","Text":"It\u0027s cosine squared x all squared,"},{"Start":"01:05.760 ","End":"01:07.845","Text":"that\u0027s cosine to the 4th already,"},{"Start":"01:07.845 ","End":"01:13.760","Text":"throw in another cosine and I\u0027m back to 5 and that\u0027s all right."},{"Start":"01:13.760 ","End":"01:19.660","Text":"Now, I\u0027ll use this formula here to write this as the integral."},{"Start":"01:19.660 ","End":"01:21.130","Text":"Instead of cosine squared,"},{"Start":"01:21.130 ","End":"01:24.690","Text":"I can write 1 minus sine squared,"},{"Start":"01:24.690 ","End":"01:30.550","Text":"squared also times cosine x dx."},{"Start":"01:30.650 ","End":"01:32.885","Text":"We have what we wanted,"},{"Start":"01:32.885 ","End":"01:37.910","Text":"a function of sine x times the cosine x and when we have this case,"},{"Start":"01:37.910 ","End":"01:42.970","Text":"we substitute t equals sine x."},{"Start":"01:42.970 ","End":"01:50.160","Text":"As usual dt derivative of this cosine of x but dx."},{"Start":"01:50.160 ","End":"01:52.860","Text":"This last part is cosine x dx,"},{"Start":"01:52.860 ","End":"01:55.610","Text":"it\u0027s exactly dt and in wherever we see sine x,"},{"Start":"01:55.610 ","End":"02:03.220","Text":"we put t. That will give us 1 minus t squared, squared dt."},{"Start":"02:03.220 ","End":"02:07.040","Text":"Now, some algebra, just expand this and we get the"},{"Start":"02:07.040 ","End":"02:11.825","Text":"integral a minus b squared is a squared minus 2ab plus b squared."},{"Start":"02:11.825 ","End":"02:22.160","Text":"It\u0027s 1 squared minus 2t squared plus t^4th dt,"},{"Start":"02:22.160 ","End":"02:24.214","Text":"which is a straightforward integral."},{"Start":"02:24.214 ","End":"02:31.520","Text":"It\u0027s t minus 2/3t^3 plus t^5th over"},{"Start":"02:31.520 ","End":"02:35.445","Text":"5 plus a constant"},{"Start":"02:35.445 ","End":"02:42.440","Text":"and all that remains now is to substitute back so we get from t to x."},{"Start":"02:42.440 ","End":"02:46.385","Text":"Using this original substitution,"},{"Start":"02:46.385 ","End":"02:54.710","Text":"put sine x instead of t. Sine x minus 2/3 sine"},{"Start":"02:54.710 ","End":"03:03.900","Text":"cubed x plus 1/5 sine to the 5th x plus a constant."},{"Start":"03:03.900 ","End":"03:09.255","Text":"That\u0027s part a. Here I copied part b."},{"Start":"03:09.255 ","End":"03:13.700","Text":"What I\u0027d like to do is use the same trick again as before"},{"Start":"03:13.700 ","End":"03:18.350","Text":"and look for a way to write this as a function of sine x times cosine x,"},{"Start":"03:18.350 ","End":"03:19.775","Text":"so the other way around."},{"Start":"03:19.775 ","End":"03:22.655","Text":"But here I don\u0027t see any sine or cosine."},{"Start":"03:22.655 ","End":"03:27.395","Text":"Well, that\u0027s easily fixed if we remember that the tangent is sine over cosine,"},{"Start":"03:27.395 ","End":"03:30.950","Text":"then we start off with integral of sine to"},{"Start":"03:30.950 ","End":"03:38.190","Text":"the 5th x over cosine to the 5th x dx."},{"Start":"03:38.190 ","End":"03:42.620","Text":"Which one will I go with something times sine x or something times cosine x?"},{"Start":"03:42.620 ","End":"03:45.530","Text":"Well, I think we\u0027ll go with something times sine x because"},{"Start":"03:45.530 ","End":"03:48.605","Text":"I already had an extra sine x in the numerator."},{"Start":"03:48.605 ","End":"03:51.650","Text":"This is an odd number and then I\u0027ll have sine to the 4th and when I have an"},{"Start":"03:51.650 ","End":"03:55.190","Text":"even it\u0027s easy to convert from sine to cosine."},{"Start":"03:55.190 ","End":"03:58.310","Text":"Well, this is partly from experience,"},{"Start":"03:58.310 ","End":"04:00.020","Text":"but let\u0027s try it that way."},{"Start":"04:00.020 ","End":"04:06.710","Text":"Let\u0027s write it as the integral of sine to the 4th but I\u0027ll do two steps in one."},{"Start":"04:06.710 ","End":"04:08.270","Text":"Instead of sine to the 4th,"},{"Start":"04:08.270 ","End":"04:11.105","Text":"I\u0027ll do sine squared, squared,"},{"Start":"04:11.105 ","End":"04:17.955","Text":"that\u0027s sine to the 4th over cosine to the 5th x."},{"Start":"04:17.955 ","End":"04:19.950","Text":"Now, I\u0027m still missing a sine x because this"},{"Start":"04:19.950 ","End":"04:22.320","Text":"only give me to the 4th and I need to the 5th."},{"Start":"04:22.320 ","End":"04:30.110","Text":"Here, sine x dx and now there\u0027s a 3rd formula missing to this."},{"Start":"04:30.110 ","End":"04:33.200","Text":"We also have that the sine squared of Alpha is"},{"Start":"04:33.200 ","End":"04:38.630","Text":"1 minus cosine squared of Alpha which I\u0027ll use here with Alpha equals x."},{"Start":"04:38.630 ","End":"04:47.435","Text":"What we get is the integral of 1 minus cosine squared x"},{"Start":"04:47.435 ","End":"04:52.300","Text":"squared over cosine to"},{"Start":"04:52.300 ","End":"04:59.580","Text":"the 5th x times sine x dx."},{"Start":"04:59.580 ","End":"05:03.980","Text":"Good. We\u0027ve got it into the form of a function of cosine x times"},{"Start":"05:03.980 ","End":"05:05.930","Text":"the sine x and then we know that"},{"Start":"05:05.930 ","End":"05:11.145","Text":"the substitution t equals cosine x is going to work for us."},{"Start":"05:11.145 ","End":"05:18.305","Text":"As before, dt is minus sine x dx,"},{"Start":"05:18.305 ","End":"05:22.130","Text":"which means that when we get to the sine x dx that\u0027s here,"},{"Start":"05:22.130 ","End":"05:26.070","Text":"we\u0027ll have to remember that that\u0027s minus dt."},{"Start":"05:26.070 ","End":"05:28.785","Text":"Just throwing a minus around no big deal."},{"Start":"05:28.785 ","End":"05:35.555","Text":"So make the substitution and we get 1 minus t squared,"},{"Start":"05:35.555 ","End":"05:42.520","Text":"squared over t^5th times minus dt."},{"Start":"05:42.520 ","End":"05:44.375","Text":"I said we have a minus here."},{"Start":"05:44.375 ","End":"05:48.305","Text":"Let\u0027s put the minus in front and the dt here."},{"Start":"05:48.305 ","End":"05:54.980","Text":"Expanding we get minus the integral of this thing"},{"Start":"05:54.980 ","End":"06:02.390","Text":"squared is 1 squared minus twice one times t squared plus t squared,"},{"Start":"06:02.390 ","End":"06:06.440","Text":"squared all over t^5th."},{"Start":"06:06.440 ","End":"06:10.350","Text":"A little bit more algebra."},{"Start":"06:10.350 ","End":"06:15.375","Text":"Ignore this 1 over t^5th is a 5 here."},{"Start":"06:15.375 ","End":"06:24.450","Text":"1 over t^5th or t to the minus 5 if you like minus 2t squared over t^5th,"},{"Start":"06:24.450 ","End":"06:27.210","Text":"2 minus 5 is minus 3,"},{"Start":"06:27.210 ","End":"06:37.770","Text":"so it\u0027s minus 2t to the minus 3 and the last one t^4th over t^5th is t^4 minus 5,"},{"Start":"06:37.770 ","End":"06:43.000","Text":"t to the minus 1, all this dt."},{"Start":"06:43.040 ","End":"06:46.760","Text":"Now, we have exponents powers of t,"},{"Start":"06:46.760 ","End":"06:48.815","Text":"and we can easily do those."},{"Start":"06:48.815 ","End":"06:52.169","Text":"We get minus the integral."},{"Start":"06:52.169 ","End":"06:55.275","Text":"We raise the power by 1 and divide by it,"},{"Start":"06:55.275 ","End":"06:59.100","Text":"so t to the minus 4 over minus 4,"},{"Start":"06:59.100 ","End":"07:04.995","Text":"it\u0027s minus 1/4t to the minus 4,"},{"Start":"07:04.995 ","End":"07:09.425","Text":"and here raise it by 1, it\u0027s minus 2."},{"Start":"07:09.425 ","End":"07:12.710","Text":"Minus 2 over minus 2 is plus 1."},{"Start":"07:12.710 ","End":"07:17.905","Text":"It\u0027s plus t to the minus 2,"},{"Start":"07:17.905 ","End":"07:22.790","Text":"and then we can\u0027t do that rule because it\u0027s a minus 1."},{"Start":"07:22.790 ","End":"07:26.960","Text":"This is the natural logarithm plus"},{"Start":"07:26.960 ","End":"07:35.030","Text":"natural log of absolute value of t. Of course,"},{"Start":"07:35.030 ","End":"07:37.370","Text":"this is not their integral, excuse me."},{"Start":"07:37.370 ","End":"07:38.850","Text":"I mean, this is the integral,"},{"Start":"07:38.850 ","End":"07:42.860","Text":"just need regular brackets plus a constant."},{"Start":"07:42.860 ","End":"07:48.545","Text":"Then finally, what I do is I need to get back to the world of x from the world of t,"},{"Start":"07:48.545 ","End":"07:51.070","Text":"do the reverse substitution."},{"Start":"07:51.070 ","End":"07:54.800","Text":"We have, let\u0027s see, minus, minus is plus,"},{"Start":"07:54.800 ","End":"07:58.430","Text":"so it\u0027s 1/4 and t is"},{"Start":"07:58.430 ","End":"08:07.520","Text":"cosine x. Cosine x to the minus 4 is 1 over cosine to the 4th x,"},{"Start":"08:07.520 ","End":"08:10.395","Text":"t to the minus 2 is 1 over t squared,"},{"Start":"08:10.395 ","End":"08:14.080","Text":"1 over cosine squared x."},{"Start":"08:14.870 ","End":"08:17.785","Text":"Hang on, there\u0027s a minus out here,"},{"Start":"08:17.785 ","End":"08:21.040","Text":"minus out here that makes this minus."},{"Start":"08:21.040 ","End":"08:24.520","Text":"In the last one, we\u0027ll also have a minus because we had the minus here,"},{"Start":"08:24.520 ","End":"08:31.690","Text":"so we have minus natural log of absolute value"},{"Start":"08:31.690 ","End":"08:39.630","Text":"of cosine x because that\u0027s what t is and plus the constant,"},{"Start":"08:39.630 ","End":"08:43.185","Text":"and we\u0027re done with part b."},{"Start":"08:43.185 ","End":"08:47.445","Text":"Here\u0027s part c, which I copied from here."},{"Start":"08:47.445 ","End":"08:53.705","Text":"Again, I\u0027d like to try and get it into the form of a function of sine x times cosine x,"},{"Start":"08:53.705 ","End":"08:56.090","Text":"or the other way round function of cosine x times"},{"Start":"08:56.090 ","End":"08:59.885","Text":"sine x. I don\u0027t have anything in the numerator at all."},{"Start":"08:59.885 ","End":"09:03.920","Text":"I had the idea that if I multiply top and bottom by cosine x,"},{"Start":"09:03.920 ","End":"09:05.150","Text":"that might help me."},{"Start":"09:05.150 ","End":"09:06.500","Text":"Let\u0027s try doing that."},{"Start":"09:06.500 ","End":"09:09.010","Text":"I put cosine x in the top,"},{"Start":"09:09.010 ","End":"09:11.730","Text":"I\u0027ll get cosine x,"},{"Start":"09:11.730 ","End":"09:14.495","Text":"and if I multiply the bottom by cosine x,"},{"Start":"09:14.495 ","End":"09:17.410","Text":"I\u0027ll just get cosine squared x."},{"Start":"09:17.410 ","End":"09:21.980","Text":"Now, this looks good to me because this could be my cosine x dx and this I"},{"Start":"09:21.980 ","End":"09:26.530","Text":"can convert into sine x because we\u0027ve done this before."},{"Start":"09:26.530 ","End":"09:33.740","Text":"What we get is cosine squared is 1 minus sine squared."},{"Start":"09:33.740 ","End":"09:39.605","Text":"Let me write the 1 over 1 minus sine squared separately."},{"Start":"09:39.605 ","End":"09:42.590","Text":"That takes care of this cosine squared in the bottom."},{"Start":"09:42.590 ","End":"09:44.150","Text":"Then this cosine x,"},{"Start":"09:44.150 ","End":"09:46.145","Text":"I\u0027ll put it the side here,"},{"Start":"09:46.145 ","End":"09:48.710","Text":"followed by the dx."},{"Start":"09:48.710 ","End":"09:53.800","Text":"Now, this is very good because I have my function of sine x"},{"Start":"09:53.800 ","End":"09:59.200","Text":"here followed by cosine x and I know that in this case,"},{"Start":"09:59.200 ","End":"10:02.670","Text":"the substitution t equals sine x is going to work."},{"Start":"10:02.670 ","End":"10:08.650","Text":"Here we are, t equals sine x and we also need dt,"},{"Start":"10:08.650 ","End":"10:15.850","Text":"this is very familiar to us already is cosine x dx and"},{"Start":"10:15.850 ","End":"10:23.440","Text":"the substitution gives us the integral of 1 over 1 minus t squared."},{"Start":"10:23.440 ","End":"10:28.820","Text":"That\u0027s from sine x being t and the cosine x dx is dt."},{"Start":"10:28.860 ","End":"10:33.365","Text":"Here we have this integral and how do we solve this?"},{"Start":"10:33.365 ","End":"10:37.880","Text":"Well, there is a method called decomposition into partial fractions,"},{"Start":"10:37.880 ","End":"10:40.940","Text":"and there\u0027s a chapter on that but I\u0027m not going to do that"},{"Start":"10:40.940 ","End":"10:44.600","Text":"here because in most formula sheets this will be on it"},{"Start":"10:44.600 ","End":"10:52.895","Text":"and I\u0027ll just simply quote the result that this is equal to 1/2 the natural log"},{"Start":"10:52.895 ","End":"11:03.015","Text":"of 1 minus t over 1 plus t plus a constant if I\u0027m not mistaken."},{"Start":"11:03.015 ","End":"11:07.879","Text":"Now I just have to plug in t to get back to x,"},{"Start":"11:07.879 ","End":"11:09.410","Text":"so t is sine x,"},{"Start":"11:09.410 ","End":"11:15.070","Text":"so we get 1/2 natural log of"},{"Start":"11:15.070 ","End":"11:25.355","Text":"1 minus sine x over 1 plus sine x in absolute value plus C,"},{"Start":"11:25.355 ","End":"11:29.970","Text":"and that does part c and so we\u0027re done."}],"ID":6726},{"Watched":false,"Name":"Exercise 4","Duration":"15m 37s","ChapterTopicVideoID":6666,"CourseChapterTopicPlaylistID":1607,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we have to compute the following 3 integrals."},{"Start":"00:04.740 ","End":"00:08.235","Text":"I\u0027ve copied the first one over here."},{"Start":"00:08.235 ","End":"00:11.610","Text":"As usual, what I\u0027m going to try and do is get"},{"Start":"00:11.610 ","End":"00:15.495","Text":"this as a function of sine x times cosine x,"},{"Start":"00:15.495 ","End":"00:18.615","Text":"or the other way around, the function of cosine x times sine x."},{"Start":"00:18.615 ","End":"00:22.620","Text":"We had a similar one like this in another exercise with cosine x."},{"Start":"00:22.620 ","End":"00:27.420","Text":"It looks like the way to go is to multiply top and bottom by sine x,"},{"Start":"00:27.420 ","End":"00:31.995","Text":"and then I get that this is equal to the integral of"},{"Start":"00:31.995 ","End":"00:38.640","Text":"sine x over sine squared x dx."},{"Start":"00:38.640 ","End":"00:41.295","Text":"Just multiplying top and bottom by sine x."},{"Start":"00:41.295 ","End":"00:43.570","Text":"Here I have my sine x dx."},{"Start":"00:43.570 ","End":"00:47.380","Text":"This sine squared could easily be converted into cosines."},{"Start":"00:47.380 ","End":"00:53.480","Text":"I usually write this formula with me because from here I can get to the other 2."},{"Start":"00:53.480 ","End":"00:56.315","Text":"If I want sine squared in terms of cosine squared,"},{"Start":"00:56.315 ","End":"01:00.530","Text":"I just throw the cosine squared to the other side and I get sine"},{"Start":"01:00.530 ","End":"01:08.985","Text":"squared of Alpha is 1 minus cosine squared Alpha."},{"Start":"01:08.985 ","End":"01:13.520","Text":"What I get here is the integral of"},{"Start":"01:13.520 ","End":"01:17.165","Text":"1 over 1 minus cosine"},{"Start":"01:17.165 ","End":"01:24.135","Text":"squared x times sine x dx."},{"Start":"01:24.135 ","End":"01:28.930","Text":"Now I have a function of cosine x times sine x."},{"Start":"01:28.930 ","End":"01:30.575","Text":"We know how to handle this."},{"Start":"01:30.575 ","End":"01:35.125","Text":"The substitution where t equals cosine x, we\u0027ll do it."},{"Start":"01:35.125 ","End":"01:44.095","Text":"T equals cosine x and dt is minus sine x dx."},{"Start":"01:44.095 ","End":"01:47.180","Text":"Substituting, we get the integral."},{"Start":"01:47.180 ","End":"01:48.650","Text":"Since cosine x is t,"},{"Start":"01:48.650 ","End":"01:53.540","Text":"we have 1 over 1 minus t squared."},{"Start":"01:53.540 ","End":"01:58.500","Text":"Sine x dx is going to be just this part."},{"Start":"01:58.500 ","End":"02:01.865","Text":"The minus we\u0027ll throw on the other side minus dt."},{"Start":"02:01.865 ","End":"02:05.920","Text":"Here\u0027s the dt and the minus I\u0027ll put in front."},{"Start":"02:05.920 ","End":"02:09.334","Text":"Then I can write it as the integral of 1 over,"},{"Start":"02:09.334 ","End":"02:11.600","Text":"just reverse the order because of the minus,"},{"Start":"02:11.600 ","End":"02:14.570","Text":"t squared minus 1 dt."},{"Start":"02:14.570 ","End":"02:17.225","Text":"This is the hard integral to solve."},{"Start":"02:17.225 ","End":"02:22.940","Text":"Normal way of doing it is with reduction to partial fractions."},{"Start":"02:22.940 ","End":"02:25.580","Text":"But there\u0027s a chapter on that and you can go and check it out,"},{"Start":"02:25.580 ","End":"02:29.599","Text":"there is also formula sheets that contain this integral."},{"Start":"02:29.599 ","End":"02:35.930","Text":"I\u0027ll just give it you from the formula sheet that this is equal to one-half times"},{"Start":"02:35.930 ","End":"02:45.805","Text":"the natural logarithm of absolute value of t minus 1 over t plus 1."},{"Start":"02:45.805 ","End":"02:48.855","Text":"Yeah, plus the constant."},{"Start":"02:48.855 ","End":"02:53.870","Text":"All I have to do now is get back to the world of x from"},{"Start":"02:53.870 ","End":"02:59.345","Text":"the world of t by substituting back t equals cosine x."},{"Start":"02:59.345 ","End":"03:06.035","Text":"I have one-half natural logarithm of cosine x"},{"Start":"03:06.035 ","End":"03:13.325","Text":"minus 1 over cosine x plus 1 and plus the constant."},{"Start":"03:13.325 ","End":"03:15.955","Text":"Okay, that\u0027s it, part a."},{"Start":"03:15.955 ","End":"03:19.050","Text":"Here\u0027s part b over here"},{"Start":"03:19.050 ","End":"03:22.940","Text":"and I\u0027ve written the formula here that will be useful, you\u0027ll see."},{"Start":"03:22.940 ","End":"03:29.510","Text":"Again what I want to do is write a function of cosine x times sine x or as"},{"Start":"03:29.510 ","End":"03:37.035","Text":"a function of sine x times cosine x. I can do this if I expand sine 2x."},{"Start":"03:37.035 ","End":"03:43.340","Text":"I can write this as the integral of 2 sine x"},{"Start":"03:43.340 ","End":"03:52.315","Text":"cosine x times e to the power of cosine x dx."},{"Start":"03:52.315 ","End":"03:55.130","Text":"Now, this is from this formula here."},{"Start":"03:55.130 ","End":"03:59.600","Text":"Now a small rearrangement will give me that this is equal 2,"},{"Start":"03:59.600 ","End":"04:02.755","Text":"let\u0027s see the 2 I can take outside,"},{"Start":"04:02.755 ","End":"04:04.485","Text":"I can get the integral."},{"Start":"04:04.485 ","End":"04:07.325","Text":"I\u0027ll first of all, write the cosine of x,"},{"Start":"04:07.325 ","End":"04:11.570","Text":"then e to the cosine x,"},{"Start":"04:11.570 ","End":"04:18.210","Text":"and then the sine x dx."},{"Start":"04:18.210 ","End":"04:19.895","Text":"Now I have what I want,"},{"Start":"04:19.895 ","End":"04:24.995","Text":"I have a function of cosine x here in green and I have alongside it the sine x,"},{"Start":"04:24.995 ","End":"04:32.085","Text":"which means that I should make the substitution t equals cosine x."},{"Start":"04:32.085 ","End":"04:35.805","Text":"Along with this, dt equals,"},{"Start":"04:35.805 ","End":"04:40.830","Text":"but it\u0027s minus sine x dx."},{"Start":"04:40.830 ","End":"04:46.995","Text":"I\u0027ve got to remember that when I substitute this sine x dx is going to be minus dt,"},{"Start":"04:46.995 ","End":"04:49.005","Text":"because this minus will go here."},{"Start":"04:49.005 ","End":"04:52.580","Text":"What I\u0027ll get, in fact the minus I\u0027ll put in the beginning,"},{"Start":"04:52.580 ","End":"04:54.680","Text":"you don\u0027t have to worry about it now."},{"Start":"04:54.680 ","End":"04:57.815","Text":"Integral cosine x is t,"},{"Start":"04:57.815 ","End":"05:02.890","Text":"e to the t sine x dx is minus dt,"},{"Start":"05:02.890 ","End":"05:06.810","Text":"but we already took care of the minus, so dt."},{"Start":"05:07.300 ","End":"05:10.745","Text":"Another question is how to solve this integral."},{"Start":"05:10.745 ","End":"05:14.390","Text":"Well, I suggest doing it by parts."},{"Start":"05:14.390 ","End":"05:18.635","Text":"Let me remind you what integration by part says."},{"Start":"05:18.635 ","End":"05:24.260","Text":"It says that if we have the integral of udv,"},{"Start":"05:24.260 ","End":"05:33.330","Text":"it\u0027s equal to uv minus the integral of vdu."},{"Start":"05:33.330 ","End":"05:35.840","Text":"Again, we have to do some substitutions,"},{"Start":"05:35.840 ","End":"05:39.094","Text":"I have to say which is u and which is dv,"},{"Start":"05:39.094 ","End":"05:41.645","Text":"it\u0027s easiest to take u as the polynomial."},{"Start":"05:41.645 ","End":"05:42.950","Text":"Well, at least in this case,"},{"Start":"05:42.950 ","End":"05:48.470","Text":"I can see that the integral of t e to the power of t dt."},{"Start":"05:48.470 ","End":"05:52.179","Text":"I\u0027m going to let this bit be u,"},{"Start":"05:52.179 ","End":"05:57.150","Text":"this e to the tdt will be dv."},{"Start":"05:57.150 ","End":"06:00.129","Text":"This has got to equal"},{"Start":"06:01.280 ","End":"06:08.970","Text":"uv minus the integral of vdu."},{"Start":"06:08.970 ","End":"06:13.810","Text":"I have to see what\u0027s v and what is du."},{"Start":"06:14.540 ","End":"06:17.080","Text":"Okay, well, let\u0027s see."},{"Start":"06:17.080 ","End":"06:28.150","Text":"If u is equal to t and we have also that dv is e to the tdt."},{"Start":"06:28.170 ","End":"06:32.440","Text":"I\u0027m writing it like this because there\u0027s 4 quantities I need and I have two of them,"},{"Start":"06:32.440 ","End":"06:34.075","Text":"I have u, I have dv,"},{"Start":"06:34.075 ","End":"06:36.920","Text":"I need to know what is du,"},{"Start":"06:36.920 ","End":"06:40.540","Text":"and I need to know what v is."},{"Start":"06:40.540 ","End":"06:44.710","Text":"Well, if u is t, if I differentiate this,"},{"Start":"06:44.710 ","End":"06:48.170","Text":"I get 1du equals 1dt."},{"Start":"06:49.020 ","End":"06:51.574","Text":"Here I got to going to do the opposite."},{"Start":"06:51.574 ","End":"06:54.890","Text":"I have already supposedly the derivative of something,"},{"Start":"06:54.890 ","End":"06:58.680","Text":"I need its primitive to integrate this."},{"Start":"06:58.680 ","End":"07:04.425","Text":"V is equal to also e to the t. Now I can plug these bits in."},{"Start":"07:04.425 ","End":"07:07.589","Text":"I have u, which is t,"},{"Start":"07:07.589 ","End":"07:11.920","Text":"v which is e to the t."},{"Start":"07:11.940 ","End":"07:19.970","Text":"v again is e^t and du is dt."},{"Start":"07:20.270 ","End":"07:25.710","Text":"All I have left now to do is the integral of e^t dt."},{"Start":"07:25.710 ","End":"07:28.605","Text":"Let me just clear up some of this stuff."},{"Start":"07:28.605 ","End":"07:38.770","Text":"We already have the t e^t minus the integral of e^t is just e^t."},{"Start":"07:38.770 ","End":"07:40.780","Text":"We\u0027re not putting in the plus constant yet,"},{"Start":"07:40.780 ","End":"07:42.730","Text":"we do that 1 time at the end."},{"Start":"07:42.730 ","End":"07:45.265","Text":"Getting back to here,"},{"Start":"07:45.265 ","End":"07:48.730","Text":"what I have from the side exercises minus"},{"Start":"07:48.730 ","End":"07:59.065","Text":"2 times t e^t minus e^t plus constant."},{"Start":"07:59.065 ","End":"08:01.615","Text":"We\u0027ve done the integration by parts."},{"Start":"08:01.615 ","End":"08:05.605","Text":"Now we\u0027re still in the substitution where we have to go back,"},{"Start":"08:05.605 ","End":"08:09.430","Text":"do the reverse substitution so we can get back to x."},{"Start":"08:09.430 ","End":"08:19.460","Text":"We just take the t equals cosine x and write this as minus 2 cosine x,"},{"Start":"08:19.470 ","End":"08:28.495","Text":"e^cosinex minus e^cosine x plus constant."},{"Start":"08:28.495 ","End":"08:31.555","Text":"That\u0027s the answer to part B."},{"Start":"08:31.555 ","End":"08:34.885","Text":"Though we\u0027re done, you might want to just tidy up a bit."},{"Start":"08:34.885 ","End":"08:39.490","Text":"Could take say, e^cosine x outside the brackets and get minus"},{"Start":"08:39.490 ","End":"08:47.140","Text":"2 e^cosine x times cosine x minus 1."},{"Start":"08:47.140 ","End":"08:52.520","Text":"A bit tidier. Onto the next part."},{"Start":"08:52.860 ","End":"08:56.890","Text":"Here I have part C copied."},{"Start":"08:56.890 ","End":"09:00.235","Text":"As usual, I want to try and get it as some function of"},{"Start":"09:00.235 ","End":"09:03.595","Text":"cosine x and then at the side sine x,"},{"Start":"09:03.595 ","End":"09:06.860","Text":"or the other way around with sine and cosine."},{"Start":"09:06.900 ","End":"09:11.470","Text":"First thing I notice is that everything here is sine x or cosine x,"},{"Start":"09:11.470 ","End":"09:13.150","Text":"but this is not cosine x,"},{"Start":"09:13.150 ","End":"09:15.235","Text":"this is cosine 2x."},{"Start":"09:15.235 ","End":"09:19.690","Text":"What I\u0027d like to do is keep this sine x at the side and try and"},{"Start":"09:19.690 ","End":"09:24.925","Text":"convert this all to the form of something with cosine x without the 2x."},{"Start":"09:24.925 ","End":"09:30.610","Text":"Now remember, there are actually 3 formulas for cosine 2Alpha and this is 1 of the 3."},{"Start":"09:30.610 ","End":"09:32.560","Text":"There\u0027s cosine squared minus sine squared,"},{"Start":"09:32.560 ","End":"09:35.260","Text":"there\u0027s 1 minus 2 sine squared and there\u0027s this 1."},{"Start":"09:35.260 ","End":"09:39.620","Text":"This is the 1 I\u0027ll take because it gives me everything in terms of the cosine."},{"Start":"09:39.810 ","End":"09:50.140","Text":"I\u0027m going to rewrite this as the integral of 2 over."},{"Start":"09:50.140 ","End":"09:53.470","Text":"I\u0027m going to put the sine x dx at the side."},{"Start":"09:53.470 ","End":"09:55.375","Text":"I\u0027m not putting the sine x yet."},{"Start":"09:55.375 ","End":"10:03.280","Text":"The cosine 2x from here is 2 cosine squared x minus 1."},{"Start":"10:03.280 ","End":"10:10.390","Text":"Then plus 4 cosine x plus 7."},{"Start":"10:10.390 ","End":"10:15.835","Text":"Then at the side, the sine x together with the dx."},{"Start":"10:15.835 ","End":"10:19.840","Text":"This is exactly what I would like to have,"},{"Start":"10:19.840 ","End":"10:23.800","Text":"because now I have a function of cosine x and only cosine x,"},{"Start":"10:23.800 ","End":"10:27.055","Text":"no 2x\u0027s and sine x at the side."},{"Start":"10:27.055 ","End":"10:35.515","Text":"Which means that the substitution where t equals cosine x is going to work."},{"Start":"10:35.515 ","End":"10:44.515","Text":"From here dt is equal to minus sine x dx."},{"Start":"10:44.515 ","End":"10:47.380","Text":"We can start substituting."},{"Start":"10:47.380 ","End":"10:55.345","Text":"What we get is the integral of 2 over,"},{"Start":"10:55.345 ","End":"10:56.950","Text":"wherever I see cosine x,"},{"Start":"10:56.950 ","End":"11:00.775","Text":"I put t. This is 2t squared."},{"Start":"11:00.775 ","End":"11:03.865","Text":"I\u0027ll put the minus 1 with the plus 7 together."},{"Start":"11:03.865 ","End":"11:06.685","Text":"I have plus 4t from there."},{"Start":"11:06.685 ","End":"11:09.730","Text":"Minus 1 plus 7 is 6."},{"Start":"11:09.730 ","End":"11:16.120","Text":"Here sine x dx is not exactly dt."},{"Start":"11:16.120 ","End":"11:20.305","Text":"There\u0027s a minus here which I can bring to the other side, it\u0027s minus dt."},{"Start":"11:20.305 ","End":"11:25.930","Text":"Minus dt, I\u0027ll write the dt here and the minus say here."},{"Start":"11:25.930 ","End":"11:29.110","Text":"1 more simplification."},{"Start":"11:29.110 ","End":"11:37.900","Text":"We\u0027ll just divide top and bottom by 2 and we\u0027ll get minus 1 over t"},{"Start":"11:37.900 ","End":"11:46.630","Text":"squared plus 2t plus 3 dt."},{"Start":"11:46.630 ","End":"11:50.065","Text":"This is not an easy 1 to solve."},{"Start":"11:50.065 ","End":"11:53.875","Text":"In fact, we should really use the formula sheets."},{"Start":"11:53.875 ","End":"12:00.490","Text":"There is a formula for when I have on the denominator t squared plus something squared."},{"Start":"12:00.490 ","End":"12:02.260","Text":"But it\u0027s not quite like that here."},{"Start":"12:02.260 ","End":"12:03.670","Text":"There\u0027s a middle thing."},{"Start":"12:03.670 ","End":"12:07.210","Text":"I\u0027m going to use a technique called completing the square."},{"Start":"12:07.210 ","End":"12:10.330","Text":"Let me do that at the side."},{"Start":"12:10.330 ","End":"12:18.875","Text":"We\u0027ll start off with t squared plus 2t plus 3."},{"Start":"12:18.875 ","End":"12:25.570","Text":"I want this to be t plus something squared."},{"Start":"12:25.570 ","End":"12:28.270","Text":"If it was going to be t plus something squared,"},{"Start":"12:28.270 ","End":"12:34.495","Text":"it would be t plus 1 squared plus or minus something else."},{"Start":"12:34.495 ","End":"12:38.215","Text":"Now t plus 1 squared is,"},{"Start":"12:38.215 ","End":"12:43.825","Text":"this is t squared plus 2t plus 1."},{"Start":"12:43.825 ","End":"12:47.785","Text":"But what I have is not plus 1, it\u0027s plus 3."},{"Start":"12:47.785 ","End":"12:51.250","Text":"If I put an extra plus 2 here,"},{"Start":"12:51.250 ","End":"12:52.480","Text":"then it will be okay."},{"Start":"12:52.480 ","End":"12:55.795","Text":"You\u0027ll have t squared plus 2t plus 1 plus 2,"},{"Start":"12:55.795 ","End":"12:57.700","Text":"and that\u0027s fine now."},{"Start":"12:57.700 ","End":"13:00.385","Text":"Next thing I want to do,"},{"Start":"13:00.385 ","End":"13:02.095","Text":"this is a bit long,"},{"Start":"13:02.095 ","End":"13:08.650","Text":"but bear with me, is instead of t plus 1 to put s. I can use 1 of the standard formulas."},{"Start":"13:08.650 ","End":"13:14.365","Text":"If I let now a second substitution where s equals t plus 1."},{"Start":"13:14.365 ","End":"13:20.575","Text":"Here, I\u0027ll get exactly ds is equal to dt because the 1 doesn\u0027t contribute anything."},{"Start":"13:20.575 ","End":"13:24.220","Text":"Now I can write this as,"},{"Start":"13:24.220 ","End":"13:26.830","Text":"I\u0027ll put the minus outside the integral,"},{"Start":"13:26.830 ","End":"13:37.660","Text":"minus the integral of 1 over s squared plus 2 ds."},{"Start":"13:37.660 ","End":"13:40.480","Text":"I\u0027m going to need a formula for this."},{"Start":"13:40.480 ","End":"13:44.695","Text":"There is a formula for the integral."},{"Start":"13:44.695 ","End":"13:46.150","Text":"I\u0027ll write that down here."},{"Start":"13:46.150 ","End":"13:51.850","Text":"The integral of 1 over s squared plus a squared,"},{"Start":"13:51.850 ","End":"13:54.460","Text":"where a is some parameter, ds."},{"Start":"13:54.460 ","End":"13:56.290","Text":"It\u0027s actually written with x, not with s,"},{"Start":"13:56.290 ","End":"14:00.520","Text":"but doesn\u0027t matter, is equal to 1/a,"},{"Start":"14:00.520 ","End":"14:10.420","Text":"arctangent of s/a plus the constant."},{"Start":"14:10.420 ","End":"14:12.100","Text":"I\u0027m not going to write that yet."},{"Start":"14:12.100 ","End":"14:21.070","Text":"In here, what I get is I let my 2 to be a squared so a is the square root of 2."},{"Start":"14:21.070 ","End":"14:25.750","Text":"What I get is minus 1/a,"},{"Start":"14:25.750 ","End":"14:28.090","Text":"which is 1 over the square root of"},{"Start":"14:28.090 ","End":"14:36.070","Text":"2 arctangent of s over again,"},{"Start":"14:36.070 ","End":"14:39.565","Text":"the square root of 2 plus the constant."},{"Start":"14:39.565 ","End":"14:42.235","Text":"Now, when I substitute back,"},{"Start":"14:42.235 ","End":"14:45.490","Text":"I don\u0027t want to go from s to t and then from t to"},{"Start":"14:45.490 ","End":"14:49.090","Text":"x. I want to go all the way from s to x in 1 step."},{"Start":"14:49.090 ","End":"14:51.100","Text":"Look, s equals t plus 1,"},{"Start":"14:51.100 ","End":"14:52.780","Text":"but t is cosine x,"},{"Start":"14:52.780 ","End":"15:03.130","Text":"so s is equal to cosine x plus 1. s is equal to cosine x plus 1,"},{"Start":"15:03.130 ","End":"15:10.610","Text":"to jump all the way back from s to x. I can write that here as"},{"Start":"15:10.610 ","End":"15:19.150","Text":"minus 1 over square root of 2 arctangent of s,"},{"Start":"15:19.150 ","End":"15:28.165","Text":"which is cosine x plus 1 over square root of 2 plus the constant."},{"Start":"15:28.165 ","End":"15:30.400","Text":"That\u0027s the answer."},{"Start":"15:30.400 ","End":"15:32.500","Text":"This completes Part C,"},{"Start":"15:32.500 ","End":"15:34.764","Text":"but there were only 3 parts,"},{"Start":"15:34.764 ","End":"15:38.310","Text":"so we\u0027re done with this whole exercise."}],"ID":6727}],"Thumbnail":null,"ID":1607},{"Name":"Integration using Trigonometric substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 (with Theory)","Duration":"4m 54s","ChapterTopicVideoID":6667,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6667.jpeg","UploadDate":"2016-07-19T20:52:46.0470000","DurationForVideoObject":"PT4M54S","Description":null,"MetaTitle":"Exercise 1 (with Theory) - Integration using Trigonometric Substitution: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Integration using Trigonometric Substitution practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/trigonometric-integrals-and-trigonometric-substitution/integration-using-trigonometric-substitution/vid6728","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"Here we have to compute the following integral."},{"Start":"00:03.255 ","End":"00:08.190","Text":"Normally what we would try to do is to substitute t"},{"Start":"00:08.190 ","End":"00:12.885","Text":"equals the square root of 4 minus x squared. I tried this."},{"Start":"00:12.885 ","End":"00:14.295","Text":"It doesn\u0027t work."},{"Start":"00:14.295 ","End":"00:16.065","Text":"When it doesn\u0027t work,"},{"Start":"00:16.065 ","End":"00:20.700","Text":"there is a special method for dealing with this thing."},{"Start":"00:20.700 ","End":"00:23.955","Text":"Specifically, when we have something"},{"Start":"00:23.955 ","End":"00:27.810","Text":"like the square root of a squared minus x squared in general."},{"Start":"00:27.810 ","End":"00:29.655","Text":"Since 4 is 2 squared,"},{"Start":"00:29.655 ","End":"00:34.260","Text":"this is what we have in our case with a is equal to 2."},{"Start":"00:34.260 ","End":"00:37.710","Text":"Now there\u0027s a standard set of equations that we use in this case,"},{"Start":"00:37.710 ","End":"00:41.255","Text":"and I\u0027ll just write them down. Here we are."},{"Start":"00:41.255 ","End":"00:48.695","Text":"The first thing is that we substitute x is equal to a sine t. That\u0027s in general."},{"Start":"00:48.695 ","End":"00:51.905","Text":"We\u0027ll get later to our case where a equals 2."},{"Start":"00:51.905 ","End":"00:56.150","Text":"The reverse, which puts t in terms of x is this."},{"Start":"00:56.150 ","End":"00:59.090","Text":"This will be useful at the end when we substitute back."},{"Start":"00:59.090 ","End":"01:02.890","Text":"The other thing is that the square root part,"},{"Start":"01:02.890 ","End":"01:05.570","Text":"by use of trigonometric identities,"},{"Start":"01:05.570 ","End":"01:07.670","Text":"we can show that it\u0027s equal to a cosine of t,"},{"Start":"01:07.670 ","End":"01:08.990","Text":"but we don\u0027t have to do it each time,"},{"Start":"01:08.990 ","End":"01:11.965","Text":"just have to remember it or have it written down."},{"Start":"01:11.965 ","End":"01:15.515","Text":"The last one is what to do for dx."},{"Start":"01:15.515 ","End":"01:17.510","Text":"It\u0027s a cosine t, dt."},{"Start":"01:17.510 ","End":"01:20.930","Text":"This is obtained simply by differentiating this."},{"Start":"01:20.930 ","End":"01:23.000","Text":"Now that we have these 3,"},{"Start":"01:23.000 ","End":"01:27.065","Text":"let\u0027s see what it spells out in our case with a equals 2."},{"Start":"01:27.065 ","End":"01:37.220","Text":"Number 1 becomes that we substitute x equals 2 sine t and later on when we need it,"},{"Start":"01:37.220 ","End":"01:43.910","Text":"if we need it, t is equal to the arc sine of x over 2."},{"Start":"01:43.910 ","End":"01:49.910","Text":"The second becomes the square root of a squared is 4."},{"Start":"01:49.910 ","End":"01:56.695","Text":"4 minus x squared is equal to 2 cosine t."},{"Start":"01:56.695 ","End":"02:05.790","Text":"The last one becomes dx is equal to 2 cosine t, dt."},{"Start":"02:06.170 ","End":"02:13.190","Text":"Let\u0027s take all these and let\u0027s go back up here and substitute and see what we get."},{"Start":"02:13.190 ","End":"02:17.345","Text":"This becomes the integral dx."},{"Start":"02:17.345 ","End":"02:18.845","Text":"I look down here,"},{"Start":"02:18.845 ","End":"02:20.870","Text":"2 cosine of t, dt."},{"Start":"02:20.870 ","End":"02:29.310","Text":"2 cosine t, I\u0027ll write the dividing sign and I\u0027ll write the dt at the side."},{"Start":"02:29.310 ","End":"02:31.964","Text":"Next thing, x squared."},{"Start":"02:31.964 ","End":"02:36.555","Text":"Well, x is 2 sine t. x squared,"},{"Start":"02:36.555 ","End":"02:37.995","Text":"if we square this,"},{"Start":"02:37.995 ","End":"02:42.629","Text":"we\u0027re going to get 2 squared sine squared t. 2 squared is 4,"},{"Start":"02:42.629 ","End":"02:49.175","Text":"sine squared t. The last thing to substitute is a square root,"},{"Start":"02:49.175 ","End":"02:51.410","Text":"and that\u0027s where we use this one here,"},{"Start":"02:51.410 ","End":"02:54.215","Text":"so the square root of 4 minus x squared is just"},{"Start":"02:54.215 ","End":"03:00.660","Text":"2 cosine t. Just"},{"Start":"03:00.660 ","End":"03:04.670","Text":"to make it really clear and also because I like playing with colors,"},{"Start":"03:04.670 ","End":"03:08.570","Text":"I\u0027ve highlighted some of the things that we said that"},{"Start":"03:08.570 ","End":"03:13.010","Text":"the dx part from here is the magenta,"},{"Start":"03:13.010 ","End":"03:16.030","Text":"it becomes 2 cosine t, dt."},{"Start":"03:16.030 ","End":"03:18.945","Text":"This becomes this, and so on."},{"Start":"03:18.945 ","End":"03:23.660","Text":"When I continue, this is a trigonometric integral with cosines and sines,"},{"Start":"03:23.660 ","End":"03:26.210","Text":"and this seems easy enough to give myself room."},{"Start":"03:26.210 ","End":"03:28.240","Text":"I\u0027m going to erase these 3."},{"Start":"03:28.240 ","End":"03:30.195","Text":"Let\u0027s see what we can simplify."},{"Start":"03:30.195 ","End":"03:36.630","Text":"The 2 cosine t here and the 2 cosine of t here cancel."},{"Start":"03:36.630 ","End":"03:42.830","Text":"I can take the 1/4 outside the integral sign,"},{"Start":"03:42.830 ","End":"03:52.225","Text":"so I get 1/4 of the integral of 1 over sine squared t dt."},{"Start":"03:52.225 ","End":"03:58.775","Text":"This is an immediate integral because the integral of 1 over sine squared is cotangent."},{"Start":"03:58.775 ","End":"04:05.730","Text":"I get 1/4 cotangent t plus a constant."},{"Start":"04:05.730 ","End":"04:12.605","Text":"But see we have the variable t and we want to go back to the world of x."},{"Start":"04:12.605 ","End":"04:17.120","Text":"What we do is, this is where we use this bit that\u0027s in brackets and"},{"Start":"04:17.120 ","End":"04:21.620","Text":"we put the t is the arc sine of x over 2."},{"Start":"04:21.620 ","End":"04:23.600","Text":"I erased that bit where a is 2,"},{"Start":"04:23.600 ","End":"04:25.625","Text":"but yeah, a here is 2."},{"Start":"04:25.625 ","End":"04:31.430","Text":"This would be 1/4 cotangent"},{"Start":"04:31.430 ","End":"04:39.590","Text":"of arc sine of x over 2 plus C. Now,"},{"Start":"04:39.590 ","End":"04:40.955","Text":"I\u0027m going to stop here,"},{"Start":"04:40.955 ","End":"04:44.540","Text":"even though it is possible to continue to simplify this and get rid of"},{"Start":"04:44.540 ","End":"04:48.860","Text":"the trigonometrical stuff and go back to square roots and so on,"},{"Start":"04:48.860 ","End":"04:51.245","Text":"but you could leave it like this,"},{"Start":"04:51.245 ","End":"04:55.050","Text":"and that\u0027s what I\u0027ll do. We\u0027re done."}],"ID":6728},{"Watched":false,"Name":"Exercise 2 (with Theory)","Duration":"4m 58s","ChapterTopicVideoID":6668,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Here we have to compute the following integral."},{"Start":"00:02.580 ","End":"00:04.575","Text":"I\u0027ve just copied it over here."},{"Start":"00:04.575 ","End":"00:08.310","Text":"The first thing you would try would be to"},{"Start":"00:08.310 ","End":"00:12.435","Text":"substitute t equals the square root of 4 plus x squared."},{"Start":"00:12.435 ","End":"00:15.345","Text":"I already tried it and it fails."},{"Start":"00:15.345 ","End":"00:17.970","Text":"Then we resort to plan B,"},{"Start":"00:17.970 ","End":"00:21.810","Text":"and there is a method for solving this square root."},{"Start":"00:21.810 ","End":"00:24.030","Text":"I\u0027ll show you what I mean."},{"Start":"00:24.030 ","End":"00:28.305","Text":"Here, the 4 plus x squared is 2 squared plus x squared,"},{"Start":"00:28.305 ","End":"00:34.890","Text":"and there is a known method for solving many of the integrals of this type."},{"Start":"00:34.890 ","End":"00:40.610","Text":"Whenever you see a squared plus x squared and regular substitution fails,"},{"Start":"00:40.610 ","End":"00:44.165","Text":"we try to seek trigonometrical substitution."},{"Start":"00:44.165 ","End":"00:47.045","Text":"Here are the 3 equations."},{"Start":"00:47.045 ","End":"00:51.115","Text":"We substitute x equals a times tangent t,"},{"Start":"00:51.115 ","End":"00:53.090","Text":"and the reverse of this,"},{"Start":"00:53.090 ","End":"00:54.500","Text":"if I put t in terms of x,"},{"Start":"00:54.500 ","End":"00:56.870","Text":"is at t is arc tangent of x over a,"},{"Start":"00:56.870 ","End":"01:00.665","Text":"we\u0027ll need this at the end when we substitute back from t to x."},{"Start":"01:00.665 ","End":"01:06.230","Text":"The second thing is that this square root of a squared plus x squared is equal"},{"Start":"01:06.230 ","End":"01:11.870","Text":"to a over cosine t. The third equation is that dx is a over cosine squared t dt."},{"Start":"01:11.870 ","End":"01:14.180","Text":"Let\u0027s see what it means in our case"},{"Start":"01:14.180 ","End":"01:18.885","Text":"because in our case we have that a is equal to 2,"},{"Start":"01:18.885 ","End":"01:20.400","Text":"because 2 squared is 4,"},{"Start":"01:20.400 ","End":"01:22.499","Text":"so let\u0027s write down 1,"},{"Start":"01:22.499 ","End":"01:25.830","Text":"2, and 3 in our case."},{"Start":"01:25.830 ","End":"01:31.580","Text":"First one says that x equals 2 times tangent of t,"},{"Start":"01:31.580 ","End":"01:40.340","Text":"and later on that t equals the arc tangent of x/2."},{"Start":"01:40.340 ","End":"01:50.565","Text":"The second one says that the square root of 2 squared plus x squared is equal to 2 over"},{"Start":"01:50.565 ","End":"01:55.260","Text":"cosine t. The third one says that the dx"},{"Start":"01:55.260 ","End":"02:02.370","Text":"equals 2 over cosine squared t dt."},{"Start":"02:02.370 ","End":"02:05.575","Text":"Now that we have all these 3,"},{"Start":"02:05.575 ","End":"02:13.265","Text":"we can use them to substitute in here."},{"Start":"02:13.265 ","End":"02:16.205","Text":"I don\u0027t need this one right away."},{"Start":"02:16.205 ","End":"02:19.995","Text":"I get the integral of 1 over."},{"Start":"02:19.995 ","End":"02:23.990","Text":"I will need the square root, so I\u0027ll need number 2,"},{"Start":"02:23.990 ","End":"02:31.170","Text":"and that will be 2 over cosine t. I will need number 3."},{"Start":"02:31.170 ","End":"02:36.570","Text":"I just need dx, which is equal to 2"},{"Start":"02:36.570 ","End":"02:43.710","Text":"over cosine squared t dt."},{"Start":"02:43.710 ","End":"02:47.165","Text":"Let me just color it to show you what I did."},{"Start":"02:47.165 ","End":"02:53.820","Text":"This turquoise cyan color is number 2, I get from this to this,"},{"Start":"02:53.820 ","End":"02:57.840","Text":"and the dx using number 3 gives me this."},{"Start":"02:57.840 ","End":"03:02.435","Text":"That\u0027s what we did, and we didn\u0027t use number 1."},{"Start":"03:02.435 ","End":"03:05.360","Text":"I want to simplify this."},{"Start":"03:05.360 ","End":"03:08.315","Text":"This equals the integral."},{"Start":"03:08.315 ","End":"03:10.730","Text":"Well, let\u0027s see, a lot of stuff will cancel."},{"Start":"03:10.730 ","End":"03:15.815","Text":"The 2 in the numerator will go with the 2 in the denominator here."},{"Start":"03:15.815 ","End":"03:21.815","Text":"If I multiply 1 over cosine t times cosine squared t,"},{"Start":"03:21.815 ","End":"03:29.520","Text":"all I\u0027ll be left with is 1 over cosine t dt."},{"Start":"03:30.310 ","End":"03:36.280","Text":"Now this is a known integral and perhaps you have a formula sheet."},{"Start":"03:36.280 ","End":"03:39.200","Text":"In any event, I\u0027ll tell you what this is."},{"Start":"03:39.200 ","End":"03:45.380","Text":"It\u0027s 1/2, the natural log of 1"},{"Start":"03:45.380 ","End":"03:52.985","Text":"plus sine t over 1 minus sine t,"},{"Start":"03:52.985 ","End":"03:55.685","Text":"and then plus the constant."},{"Start":"03:55.685 ","End":"04:01.010","Text":"But I can\u0027t stop here because I have t here and not x,"},{"Start":"04:01.010 ","End":"04:04.400","Text":"so I have to substitute for t. That\u0027s why I kept this bit."},{"Start":"04:04.400 ","End":"04:08.860","Text":"I knew I\u0027d need it, and so what we\u0027ll get is messy."},{"Start":"04:08.860 ","End":"04:13.865","Text":"I think I might need more room and start further to the left."},{"Start":"04:13.865 ","End":"04:17.930","Text":"1/2 natural log of"},{"Start":"04:17.930 ","End":"04:24.095","Text":"1 plus sine of arc tangent x over 2,"},{"Start":"04:24.095 ","End":"04:29.650","Text":"arc tangent x over 2."},{"Start":"04:29.650 ","End":"04:32.330","Text":"Then on the denominator,"},{"Start":"04:32.330 ","End":"04:35.180","Text":"same thing except with a minus,"},{"Start":"04:35.180 ","End":"04:45.135","Text":"1 minus sine of arc tangent of x over 2 plus the constant."},{"Start":"04:45.135 ","End":"04:49.625","Text":"Now, I\u0027ll tell you that it is possible to get rid of all the trigonometric stuff"},{"Start":"04:49.625 ","End":"04:54.035","Text":"by simplifying and using trigonometry of right triangles."},{"Start":"04:54.035 ","End":"04:55.565","Text":"I\u0027m not going do that."},{"Start":"04:55.565 ","End":"04:59.430","Text":"We just leave the answer like this and we\u0027re done."}],"ID":6729},{"Watched":false,"Name":"Exercise 3 (with Theory)","Duration":"4m 31s","ChapterTopicVideoID":6669,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.140","Text":"Here we have to compute this integral,"},{"Start":"00:02.140 ","End":"00:03.765","Text":"I\u0027ve copied it over here."},{"Start":"00:03.765 ","End":"00:06.930","Text":"The first thing you should try is substituting"},{"Start":"00:06.930 ","End":"00:10.380","Text":"t equals the square root of x squared minus 1."},{"Start":"00:10.380 ","End":"00:17.145","Text":"I already tried it and it fails so then we use our alternative plan."},{"Start":"00:17.145 ","End":"00:22.275","Text":"There was actually a formula that often helps when we have the square root of,"},{"Start":"00:22.275 ","End":"00:24.945","Text":"well, in our case x squared minus 1,"},{"Start":"00:24.945 ","End":"00:30.840","Text":"but in general, the square root of x squared minus a squared,"},{"Start":"00:30.840 ","End":"00:35.105","Text":"which would true in our case if a equals 1."},{"Start":"00:35.105 ","End":"00:39.705","Text":"I\u0027m going to write down the standard 3 equations to use in this case."},{"Start":"00:39.705 ","End":"00:41.714","Text":"Here\u0027s the 3 equations,"},{"Start":"00:41.714 ","End":"00:43.985","Text":"I\u0027m not going to go into them in detail."},{"Start":"00:43.985 ","End":"00:46.880","Text":"These are the standard equations to use for cases like"},{"Start":"00:46.880 ","End":"00:50.045","Text":"this where the direct substitution fails."},{"Start":"00:50.045 ","End":"00:55.715","Text":"Let me just write out what it spells in our case with a equals 1."},{"Start":"00:55.715 ","End":"01:02.640","Text":"I\u0027ll write down points number 1, 2, and 3."},{"Start":"01:02.640 ","End":"01:10.085","Text":"If a is 1, it becomes simpler than x is just equal to 1 over cosine t and"},{"Start":"01:10.085 ","End":"01:20.490","Text":"the reverse substitution is t equals the arc cosine of 1/x."},{"Start":"01:20.560 ","End":"01:28.025","Text":"This one becomes the square root of x squared minus 1 squared is 1,"},{"Start":"01:28.025 ","End":"01:36.140","Text":"is equal to 1 times tangent t. The last one becomes dx is equal"},{"Start":"01:36.140 ","End":"01:45.565","Text":"to sine t when a is 1 over cosine squared t dt."},{"Start":"01:45.565 ","End":"01:49.370","Text":"If we substitute all these things in here,"},{"Start":"01:49.370 ","End":"01:52.070","Text":"we get that first of all,"},{"Start":"01:52.070 ","End":"01:59.465","Text":"the dx from number 3 gives me the sine t over"},{"Start":"01:59.465 ","End":"02:08.270","Text":"cosine squared t and dt which I will write at the side here,"},{"Start":"02:08.270 ","End":"02:10.835","Text":"I\u0027m not sure how much space I\u0027ll need for the rest."},{"Start":"02:10.835 ","End":"02:14.090","Text":"Then I have a denominator."},{"Start":"02:14.090 ","End":"02:21.065","Text":"Tell you what, I\u0027ll just extend this denominator and let\u0027s see what else we have."},{"Start":"02:21.065 ","End":"02:25.740","Text":"I\u0027m going to use formula number 2 for this one."},{"Start":"02:25.740 ","End":"02:30.350","Text":"I\u0027ll get the x squared minus 1 under the square root sign is equal to"},{"Start":"02:30.350 ","End":"02:35.195","Text":"tangent t. I have a tangent t here,"},{"Start":"02:35.195 ","End":"02:40.085","Text":"but I still have this x squared and that\u0027s where number 1 comes in."},{"Start":"02:40.085 ","End":"02:43.644","Text":"X squared is just this thing squared,"},{"Start":"02:43.644 ","End":"02:53.120","Text":"it\u0027s 1 over cosine squared t. I\u0027m just going to color some things to make it clearer."},{"Start":"02:53.120 ","End":"02:56.960","Text":"I used number 1 here for x"},{"Start":"02:56.960 ","End":"03:00.995","Text":"squared to make it 1 over cosine squared t by just squaring this."},{"Start":"03:00.995 ","End":"03:04.850","Text":"I used number 2 to let this equals tangent t and"},{"Start":"03:04.850 ","End":"03:09.530","Text":"number 3 for the dx to equals sine t over cosine squared t dt."},{"Start":"03:09.530 ","End":"03:11.840","Text":"I need to continue."},{"Start":"03:11.840 ","End":"03:13.175","Text":"Let\u0027s see now."},{"Start":"03:13.175 ","End":"03:20.750","Text":"Well, cosine squared t will cancel with 1 over cosine squared t and what we\u0027ll"},{"Start":"03:20.750 ","End":"03:30.230","Text":"get is the integral of sine t over tangent t. But instead of tangent t,"},{"Start":"03:30.230 ","End":"03:39.180","Text":"allow me to write sine t over cosine t because that\u0027s what tangent t is and dt."},{"Start":"03:39.740 ","End":"03:42.165","Text":"This is equal to,"},{"Start":"03:42.165 ","End":"03:43.890","Text":"the sine and the sine cancel,"},{"Start":"03:43.890 ","End":"03:45.680","Text":"the cosine comes to the top."},{"Start":"03:45.680 ","End":"03:49.130","Text":"I\u0027m left with cosine of t, dt."},{"Start":"03:49.130 ","End":"03:54.050","Text":"The integral of cosine sine is an immediate one,"},{"Start":"03:54.050 ","End":"03:59.000","Text":"it\u0027s just sine t plus a constant."},{"Start":"03:59.000 ","End":"04:03.050","Text":"But I can\u0027t finish here because I still have t and"},{"Start":"04:03.050 ","End":"04:07.040","Text":"I need to put it in terms of x and that\u0027s why I kept this bit,"},{"Start":"04:07.040 ","End":"04:10.160","Text":"because t is the arc cosine of 1/x."},{"Start":"04:10.160 ","End":"04:17.435","Text":"So what I get is the sine of arc cosine"},{"Start":"04:17.435 ","End":"04:24.200","Text":"1/x plus c. Now it is possible to simplify this and"},{"Start":"04:24.200 ","End":"04:27.035","Text":"get rid of the trigonometric stuff but"},{"Start":"04:27.035 ","End":"04:32.010","Text":"that\u0027s not very important and I\u0027m not going to do it, so we\u0027re done."}],"ID":6730},{"Watched":false,"Name":"Exercise 4","Duration":"7m 23s","ChapterTopicVideoID":6670,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Here we have to compute the following integral."},{"Start":"00:02.250 ","End":"00:03.975","Text":"I\u0027ve copied it down here."},{"Start":"00:03.975 ","End":"00:08.460","Text":"The first thing to do is to try substitution of t equals the square root."},{"Start":"00:08.460 ","End":"00:12.285","Text":"You don\u0027t have to do this now because I just tried it and it failed,"},{"Start":"00:12.285 ","End":"00:15.390","Text":"so we have to have a plan B."},{"Start":"00:15.390 ","End":"00:22.350","Text":"This square root very much reminds me of a formula we had"},{"Start":"00:22.350 ","End":"00:29.900","Text":"or format where we learned how to do the integral of x squared minus a squared."},{"Start":"00:29.900 ","End":"00:33.065","Text":"Now this doesn\u0027t exactly match this,"},{"Start":"00:33.065 ","End":"00:34.865","Text":"but with a little bit of manipulation,"},{"Start":"00:34.865 ","End":"00:37.340","Text":"we can get it to look like this."},{"Start":"00:37.340 ","End":"00:44.660","Text":"If I write it as the integral and I take the 4 outside the square root sign,"},{"Start":"00:44.660 ","End":"00:48.080","Text":"the 4 comes out as 2 because 4 is 2 squared."},{"Start":"00:48.080 ","End":"00:52.515","Text":"I\u0027ll do it in 2 steps in case you\u0027re not sure."},{"Start":"00:52.515 ","End":"00:57.270","Text":"The square root of 4 times the square root and I\u0027ll take 4 out here,"},{"Start":"00:57.270 ","End":"01:00.510","Text":"so I get x squared minus a 1/4."},{"Start":"01:00.510 ","End":"01:04.295","Text":"Now, I can write this as a square root of 4 is 2,"},{"Start":"01:04.295 ","End":"01:06.950","Text":"take the 2 in front of the integral sign,"},{"Start":"01:06.950 ","End":"01:10.460","Text":"and write it as the square root of x squared and"},{"Start":"01:10.460 ","End":"01:14.265","Text":"a 1/4 is a 1/2 squared minus a 1/2 squared."},{"Start":"01:14.265 ","End":"01:19.890","Text":"Now this really does look like this because if we decide to let a equals 1/2,"},{"Start":"01:19.890 ","End":"01:23.040","Text":"that\u0027s exactly the integral except for the constant,"},{"Start":"01:23.040 ","End":"01:24.985","Text":"which doesn\u0027t make any difference."},{"Start":"01:24.985 ","End":"01:27.290","Text":"I think we\u0027ve done 1 of these before."},{"Start":"01:27.290 ","End":"01:30.655","Text":"In any event, I\u0027ll write the 3 equations again."},{"Start":"01:30.655 ","End":"01:33.800","Text":"Here they are. I\u0027m not going to go into them."},{"Start":"01:33.800 ","End":"01:38.945","Text":"I\u0027ll just interpret what they mean in our case with a equals 1/2,"},{"Start":"01:38.945 ","End":"01:40.640","Text":"so rewrite 1, 2,"},{"Start":"01:40.640 ","End":"01:44.010","Text":"and 3 with a specific a."},{"Start":"01:44.010 ","End":"01:46.675","Text":"Here we have a is 1/2,"},{"Start":"01:46.675 ","End":"01:50.210","Text":"so x equals 1 over,"},{"Start":"01:50.210 ","End":"01:52.490","Text":"and then the 2 will go into the denominator,"},{"Start":"01:52.490 ","End":"01:56.370","Text":"2 cosine t. The reverse,"},{"Start":"01:56.370 ","End":"01:59.345","Text":"which we\u0027ll use later for going back from t to x,"},{"Start":"01:59.345 ","End":"02:06.165","Text":"is t equals the arc cosine a over x,"},{"Start":"02:06.165 ","End":"02:11.040","Text":"and a is a 1/2 so that will be 1 over 2x."},{"Start":"02:11.040 ","End":"02:15.140","Text":"Number 2 will give us that the square root of"},{"Start":"02:15.140 ","End":"02:19.490","Text":"x squared minus a squared will be x squared minus a 1/4,"},{"Start":"02:19.490 ","End":"02:21.200","Text":"exactly what we have here,"},{"Start":"02:21.200 ","End":"02:30.765","Text":"and that will be 1/2 tangent t. The third equation interprets as dx equals."},{"Start":"02:30.765 ","End":"02:33.765","Text":"A is a 1/2, so I just put a 2 down here."},{"Start":"02:33.765 ","End":"02:42.610","Text":"It\u0027s sine t over 2 cosine squared t dt."},{"Start":"02:43.250 ","End":"02:46.565","Text":"In order to continue,"},{"Start":"02:46.565 ","End":"02:49.590","Text":"I need to substitute."},{"Start":"02:49.590 ","End":"02:51.920","Text":"Let\u0027s see what we can do."},{"Start":"02:51.920 ","End":"02:55.160","Text":"This is equal to twice the integral."},{"Start":"02:55.160 ","End":"02:58.280","Text":"Now, the square root of x squared minus 1/2 squared,"},{"Start":"02:58.280 ","End":"03:00.305","Text":"which is x squared minus a 1/4,"},{"Start":"03:00.305 ","End":"03:02.975","Text":"is using number 2,"},{"Start":"03:02.975 ","End":"03:05.845","Text":"1/2 of tangent t,"},{"Start":"03:05.845 ","End":"03:10.520","Text":"and dx using formula 3 is"},{"Start":"03:10.520 ","End":"03:18.730","Text":"sine t over twice cosine squared t dt."},{"Start":"03:18.730 ","End":"03:25.725","Text":"Let\u0027s see now. I can cancel this 2 with this 1/2"},{"Start":"03:25.725 ","End":"03:34.595","Text":"and I can also write tangent of t as sine t over cosine t. What we get is the integral."},{"Start":"03:34.595 ","End":"03:41.580","Text":"This 2 comes out in front of sine squared t over,"},{"Start":"03:41.580 ","End":"03:47.460","Text":"cosine times cosine squared is cosine cubed t dt,"},{"Start":"03:47.570 ","End":"03:50.640","Text":"and this is equal to 1/2."},{"Start":"03:50.640 ","End":"03:54.120","Text":"Now let\u0027s see. Sine squared t is 1 minus cosine"},{"Start":"03:54.120 ","End":"03:58.610","Text":"squared t. Got everything in terms of cosines."},{"Start":"03:58.610 ","End":"04:04.255","Text":"Now, let me break it up into 2 parts."},{"Start":"04:04.255 ","End":"04:09.405","Text":"I have 1/2 the integral,"},{"Start":"04:09.405 ","End":"04:11.040","Text":"let\u0027s put a bracket here,"},{"Start":"04:11.040 ","End":"04:16.795","Text":"of 1 over cosine cubed t dt"},{"Start":"04:16.795 ","End":"04:22.175","Text":"minus cosine squared over cosine cubed is just 1 over cosine,"},{"Start":"04:22.175 ","End":"04:28.930","Text":"so we have the integral of 1 over cosine t dt."},{"Start":"04:28.930 ","End":"04:31.745","Text":"It\u0027s getting a bit complicated."},{"Start":"04:31.745 ","End":"04:38.345","Text":"Now, these powers of 1 over cosine are familiar."},{"Start":"04:38.345 ","End":"04:43.890","Text":"I think we did them in the chapter on trigonometric integrals."},{"Start":"04:43.890 ","End":"04:47.075","Text":"In any event, I\u0027m just going to quote the results of"},{"Start":"04:47.075 ","End":"04:51.380","Text":"each of these here because I believe we have done them."},{"Start":"04:51.380 ","End":"04:53.580","Text":"We get 1/2."},{"Start":"04:53.580 ","End":"05:02.255","Text":"Now, the integral of 1 over cosine cubed is 1/2 of"},{"Start":"05:02.255 ","End":"05:12.365","Text":"sine t over cosine squared t plus"},{"Start":"05:12.365 ","End":"05:19.175","Text":"1/2 natural logarithm of 1 plus sine t"},{"Start":"05:19.175 ","End":"05:28.020","Text":"over cosine t. This integral of 1 over cosine t,"},{"Start":"05:28.020 ","End":"05:30.140","Text":"we\u0027ve also done before."},{"Start":"05:30.140 ","End":"05:36.685","Text":"This comes out to be the natural log of this same thing,"},{"Start":"05:36.685 ","End":"05:45.015","Text":"1 plus sine t over cosine t. Looking very nasty."},{"Start":"05:45.015 ","End":"05:46.935","Text":"Let\u0027s continue."},{"Start":"05:46.935 ","End":"05:52.970","Text":"Altogether, what I want to say is that I can collect common terms"},{"Start":"05:52.970 ","End":"06:01.250","Text":"because this I get plus a 1/2 of something minus 1 times that something."},{"Start":"06:01.250 ","End":"06:09.300","Text":"What I want to say is that 1/2 minus 1 equals minus a 1/2. That\u0027s obvious."},{"Start":"06:09.300 ","End":"06:12.030","Text":"Now I can write this,"},{"Start":"06:12.030 ","End":"06:14.460","Text":"also taking a 1/2 out."},{"Start":"06:14.460 ","End":"06:17.520","Text":"I\u0027ll get a 1/2 something minus a 1/2 of something,"},{"Start":"06:17.520 ","End":"06:19.260","Text":"so I\u0027ll get 1/4."},{"Start":"06:19.260 ","End":"06:24.420","Text":"Then I can forget about the 1/2s and then the 1/2 came out,"},{"Start":"06:24.420 ","End":"06:32.745","Text":"so it\u0027s sine t over cosine squared t. Then minus a 1/2 of this,"},{"Start":"06:32.745 ","End":"06:34.830","Text":"but the 1/2 I already took out."},{"Start":"06:34.830 ","End":"06:41.855","Text":"It\u0027s minus natural log of absolute value of 1 plus"},{"Start":"06:41.855 ","End":"06:50.295","Text":"sine t over cosine t and plus the constant,"},{"Start":"06:50.295 ","End":"06:55.400","Text":"but we\u0027re still not done because we have to go back from the world of t to"},{"Start":"06:55.400 ","End":"07:00.720","Text":"the world of x using this substitution here."},{"Start":"07:00.720 ","End":"07:05.195","Text":"The t equals arc cosine of a over x."},{"Start":"07:05.195 ","End":"07:08.340","Text":"This is getting more involved than I actually anticipated,"},{"Start":"07:08.340 ","End":"07:11.070","Text":"so I\u0027ll just tell you how to continue."},{"Start":"07:11.070 ","End":"07:16.055","Text":"Basically we use this formula to substitute this value of t here,"},{"Start":"07:16.055 ","End":"07:19.470","Text":"here, here, and here,"},{"Start":"07:19.470 ","End":"07:22.170","Text":"and this is tedious and I\u0027m not going to do that."},{"Start":"07:22.170 ","End":"07:24.310","Text":"I\u0027m going to stop here."}],"ID":6731},{"Watched":false,"Name":"Exercise 5","Duration":"4m 35s","ChapterTopicVideoID":6671,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Here we have to compute the following integral."},{"Start":"00:02.250 ","End":"00:04.170","Text":"Just copied it over here."},{"Start":"00:04.170 ","End":"00:05.880","Text":"The first thing to try would be"},{"Start":"00:05.880 ","End":"00:09.650","Text":"the substitution t equals the square root of 4 minus x squared,"},{"Start":"00:09.650 ","End":"00:12.720","Text":"but I\u0027ve already tried it and it doesn\u0027t work."},{"Start":"00:12.720 ","End":"00:16.845","Text":"Another possibility is a trigonometric substitution."},{"Start":"00:16.845 ","End":"00:20.460","Text":"What I see here is 4 minus x squared,"},{"Start":"00:20.460 ","End":"00:24.090","Text":"which reminds me of a squared minus x squared."},{"Start":"00:24.090 ","End":"00:27.420","Text":"In fact, we\u0027ve already done one of these in a previous exercise,"},{"Start":"00:27.420 ","End":"00:30.600","Text":"and I just copied and pasted this from there."},{"Start":"00:30.600 ","End":"00:36.590","Text":"What we want to do is interpret this instead of a with a equals 2,"},{"Start":"00:36.590 ","End":"00:40.905","Text":"so we\u0027re going to get 1, 2, and 3,"},{"Start":"00:40.905 ","End":"00:47.745","Text":"x equals a sine t means x equals 2 sine t."},{"Start":"00:47.745 ","End":"00:56.330","Text":"The reverse substitution we\u0027ll need later is that t equals the arc sine of x over a,"},{"Start":"00:56.330 ","End":"00:58.055","Text":"which is x over 2."},{"Start":"00:58.055 ","End":"01:01.400","Text":"The second one says that the square root of 2 squared,"},{"Start":"01:01.400 ","End":"01:04.370","Text":"which is 4, minus x squared"},{"Start":"01:04.370 ","End":"01:11.485","Text":"is equal to a cosine t. The last one tells us what dx is,"},{"Start":"01:11.485 ","End":"01:16.530","Text":"a cosine tdt, which is 2 cosine tdt."},{"Start":"01:17.480 ","End":"01:21.180","Text":"Then we put all these in here,"},{"Start":"01:21.180 ","End":"01:26.750","Text":"so what we get is the integral x squared,"},{"Start":"01:26.750 ","End":"01:28.765","Text":"I have a formula for x,"},{"Start":"01:28.765 ","End":"01:29.935","Text":"so I just square that."},{"Start":"01:29.935 ","End":"01:31.810","Text":"Instead of 2 sine t,"},{"Start":"01:31.810 ","End":"01:39.385","Text":"I get 4 sine squared t. That\u0027s the numerator."},{"Start":"01:39.385 ","End":"01:41.980","Text":"Next bit is the denominator,"},{"Start":"01:41.980 ","End":"01:45.725","Text":"and this tells me that this is just 2 cosine t,"},{"Start":"01:45.725 ","End":"01:50.745","Text":"so here is 2 cosine t. The third bit is dx,"},{"Start":"01:50.745 ","End":"01:52.395","Text":"which I get from here,"},{"Start":"01:52.395 ","End":"01:56.740","Text":"so that\u0027s 2 cosine tdt."},{"Start":"01:57.200 ","End":"02:00.520","Text":"But I can simplify this because look,"},{"Start":"02:00.520 ","End":"02:04.555","Text":"2 cosine t and the cosine t here cancel,"},{"Start":"02:04.555 ","End":"02:10.440","Text":"so what I\u0027m left with is the integral of,"},{"Start":"02:10.440 ","End":"02:17.140","Text":"I can take the 4 out front, sine squared tdt."},{"Start":"02:17.410 ","End":"02:20.600","Text":"This is now a trigonometric integral,"},{"Start":"02:20.600 ","End":"02:23.960","Text":"and we\u0027re going to use a trick to solve it."},{"Start":"02:23.960 ","End":"02:31.400","Text":"A trigonometrical identity that I need here is that the sine of 2 Alpha,"},{"Start":"02:31.400 ","End":"02:38.535","Text":"is 1/2, 1 minus cosine 2 Alpha."},{"Start":"02:38.535 ","End":"02:40.245","Text":"What I get here,"},{"Start":"02:40.245 ","End":"02:44.505","Text":"the 4 with the 1/2 leaves me 2."},{"Start":"02:44.505 ","End":"02:47.640","Text":"Of course, instead of t, I have Alpha or the other way around."},{"Start":"02:47.640 ","End":"02:50.250","Text":"Instead of Alpha I can put t here,"},{"Start":"02:50.250 ","End":"02:53.055","Text":"sorry, I meant sine squared Alpha."},{"Start":"02:53.055 ","End":"02:56.205","Text":"Sorry, I wrote the 2 big instead of an exponent."},{"Start":"02:56.205 ","End":"03:05.230","Text":"We get 2 times the integral of 1 minus cosine 2t,"},{"Start":"03:05.230 ","End":"03:10.740","Text":"and this can split up into 2 separate integrals."},{"Start":"03:10.740 ","End":"03:17.795","Text":"What I get is 2 the integral of 1dt,"},{"Start":"03:17.795 ","End":"03:25.480","Text":"minus 2 the integral of cosine 2t dt."},{"Start":"03:25.480 ","End":"03:29.065","Text":"I just split it up according to the minus."},{"Start":"03:29.065 ","End":"03:32.365","Text":"This is equal to just,"},{"Start":"03:32.365 ","End":"03:34.490","Text":"the integral of 1 is t,"},{"Start":"03:34.490 ","End":"03:40.790","Text":"so it is 2t minus, and the integral of cosine is sine"},{"Start":"03:40.790 ","End":"03:42.695","Text":"but there\u0027s an inner derivative,"},{"Start":"03:42.695 ","End":"03:44.870","Text":"so I have to divide by 2,"},{"Start":"03:44.870 ","End":"03:46.355","Text":"and if you think about it,"},{"Start":"03:46.355 ","End":"03:49.560","Text":"the integral of this is just sine 2t."},{"Start":"03:49.560 ","End":"03:51.955","Text":"I get a 2 over 2, which cancels."},{"Start":"03:51.955 ","End":"03:53.660","Text":"Look, if you differentiate this,"},{"Start":"03:53.660 ","End":"03:56.390","Text":"you get cosine 2t times the inner derivative,"},{"Start":"03:56.390 ","End":"03:58.175","Text":"which is 2 so that looks right,"},{"Start":"03:58.175 ","End":"04:04.760","Text":"plus C. Of course we\u0027re not done because we have to substitute t here and here,"},{"Start":"04:04.760 ","End":"04:08.915","Text":"and this is where we\u0027re going to use this what I wrote here."},{"Start":"04:08.915 ","End":"04:18.665","Text":"So we get 2 arc sine of x over 2 minus"},{"Start":"04:18.665 ","End":"04:24.695","Text":"sine of 2 arc sine"},{"Start":"04:24.695 ","End":"04:30.920","Text":"of x over 2 plus C. This could be simplified,"},{"Start":"04:30.920 ","End":"04:36.300","Text":"but this is maybe not the place to do that. I\u0027m done."}],"ID":6732},{"Watched":false,"Name":"Exercise 6","Duration":"5m 35s","ChapterTopicVideoID":6672,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.110","Text":"We have to compute the following integral,"},{"Start":"00:02.110 ","End":"00:04.140","Text":"and I\u0027ve copied it over here."},{"Start":"00:04.140 ","End":"00:06.975","Text":"First thing we try is a substitution,"},{"Start":"00:06.975 ","End":"00:09.390","Text":"t equals the square root of this."},{"Start":"00:09.390 ","End":"00:11.000","Text":"I tried it, it doesn\u0027t work,"},{"Start":"00:11.000 ","End":"00:16.215","Text":"so it\u0027s probably a sign that we need a trigonometric substitution."},{"Start":"00:16.215 ","End":"00:19.500","Text":"But it doesn\u0027t fit 1 of the standard forms,"},{"Start":"00:19.500 ","End":"00:25.710","Text":"like x squared plus a squared or x squared minus a squared because there\u0027s this 2x."},{"Start":"00:25.710 ","End":"00:29.925","Text":"What we do, the usual trick is to complete the square."},{"Start":"00:29.925 ","End":"00:35.980","Text":"We essentially try to write it as x plus something squared."},{"Start":"00:36.010 ","End":"00:41.810","Text":"What I want to do is to write x plus something squared,"},{"Start":"00:41.810 ","End":"00:44.690","Text":"that will be x squared plus 2x plus something."},{"Start":"00:44.690 ","End":"00:47.600","Text":"Now if I put x plus 1 here,"},{"Start":"00:47.600 ","End":"00:49.945","Text":"I get the 1 from half of 2,"},{"Start":"00:49.945 ","End":"00:55.105","Text":"then this would give me x squared plus 2x plus 1."},{"Start":"00:55.105 ","End":"00:58.870","Text":"But if this is x squared plus 2x plus 1,"},{"Start":"00:58.870 ","End":"01:01.880","Text":"and here I have x squared plus 2x minus 3,"},{"Start":"01:01.880 ","End":"01:03.530","Text":"I have to compensate."},{"Start":"01:03.530 ","End":"01:05.720","Text":"To get from plus 1 to minus 3,"},{"Start":"01:05.720 ","End":"01:07.595","Text":"I have to subtract 4."},{"Start":"01:07.595 ","End":"01:09.800","Text":"Now this will be okay."},{"Start":"01:09.800 ","End":"01:17.590","Text":"Because now this looks very much like x squared minus a squared with a being 2."},{"Start":"01:17.590 ","End":"01:20.015","Text":"We want to try and use this,"},{"Start":"01:20.015 ","End":"01:22.370","Text":"but we don\u0027t quite have x squared."},{"Start":"01:22.370 ","End":"01:24.950","Text":"We have x plus 1 squared."},{"Start":"01:24.950 ","End":"01:28.400","Text":"a equals 2 is fine, that\u0027s 2 squared."},{"Start":"01:28.400 ","End":"01:31.550","Text":"But what to do about the x and the x plus 1?"},{"Start":"01:31.550 ","End":"01:34.445","Text":"The answer is that this works the same way."},{"Start":"01:34.445 ","End":"01:37.600","Text":"I just have to replace here x by x plus 1."},{"Start":"01:37.600 ","End":"01:44.245","Text":"I\u0027m going to erase x and write x plus 1."},{"Start":"01:44.245 ","End":"01:48.770","Text":"Then here also, I\u0027m just replacing x by x plus 1."},{"Start":"01:48.770 ","End":"01:51.980","Text":"If we differentiate this,"},{"Start":"01:51.980 ","End":"01:55.660","Text":"the derivative of x plus 1 is the same as the derivative of x,"},{"Start":"01:55.660 ","End":"01:58.040","Text":"so this formula remains unchanged."},{"Start":"01:58.040 ","End":"02:01.300","Text":"dx is this dt."},{"Start":"02:01.300 ","End":"02:06.620","Text":"Now I\u0027m going to interpret it for our case where a equals 2."},{"Start":"02:06.620 ","End":"02:09.305","Text":"We\u0027ll get 1, 2,"},{"Start":"02:09.305 ","End":"02:11.870","Text":"and 3 adjusted accordingly."},{"Start":"02:11.870 ","End":"02:16.955","Text":"So I get x plus 1 is a,"},{"Start":"02:16.955 ","End":"02:20.735","Text":"which is 2 over cosine"},{"Start":"02:20.735 ","End":"02:25.850","Text":"t. I just noticed there\u0027s 1 place I didn\u0027t substitute, sorry about that."},{"Start":"02:25.850 ","End":"02:28.100","Text":"There, I just fixed it."},{"Start":"02:28.100 ","End":"02:34.130","Text":"The second 1 says that the square root of x plus 1"},{"Start":"02:34.130 ","End":"02:42.285","Text":"squared minus 2 squared is equal to 2 tangent t,"},{"Start":"02:42.285 ","End":"02:44.575","Text":"just replacing a with 2."},{"Start":"02:44.575 ","End":"02:48.825","Text":"The last 1, dx is equal to"},{"Start":"02:48.825 ","End":"02:56.600","Text":"2 sine t over cosine squared t^dt."},{"Start":"02:56.600 ","End":"02:59.650","Text":"Now if I make all these substitutions,"},{"Start":"02:59.650 ","End":"03:02.675","Text":"I will get the integral."},{"Start":"03:02.675 ","End":"03:04.795","Text":"I forgot a dx here."},{"Start":"03:04.795 ","End":"03:07.315","Text":"Again, it\u0027s easily fixed."},{"Start":"03:07.315 ","End":"03:14.275","Text":"Now the square root of x plus 1 squared minus 4 is 2 tangent t."},{"Start":"03:14.275 ","End":"03:24.060","Text":"The dx I can get from here is 2 sine t over cosine squared t^dt."},{"Start":"03:24.060 ","End":"03:28.185","Text":"I didn\u0027t need to use this 1 this time."},{"Start":"03:28.185 ","End":"03:30.590","Text":"This is a substitution."},{"Start":"03:30.590 ","End":"03:35.030","Text":"Forgot to mention that t equals a cosine."},{"Start":"03:35.030 ","End":"03:38.650","Text":"You know what, I\u0027m going to write it down here because I\u0027ll need it later."},{"Start":"03:38.650 ","End":"03:48.335","Text":"t equals arc cosine of 2 over x plus 1."},{"Start":"03:48.335 ","End":"03:57.835","Text":"Continuing with this, this is equal to the 2 with the 2 gives me 4."},{"Start":"03:57.835 ","End":"04:01.620","Text":"Tangent is sine over cosine,"},{"Start":"04:01.620 ","End":"04:04.295","Text":"so if I put sine over cosine,"},{"Start":"04:04.295 ","End":"04:06.665","Text":"I\u0027m just increasing the powers by 1,"},{"Start":"04:06.665 ","End":"04:16.200","Text":"so I get sine squared of t over cosine cubed of t^dt."},{"Start":"04:16.220 ","End":"04:23.360","Text":"We had this in a previous exercise and it came out a real mess."},{"Start":"04:23.360 ","End":"04:26.045","Text":"I\u0027ll just quote it from the previous exercise."},{"Start":"04:26.045 ","End":"04:27.245","Text":"Well, I\u0027ll tell you what,"},{"Start":"04:27.245 ","End":"04:28.925","Text":"I\u0027m going to continue a little bit."},{"Start":"04:28.925 ","End":"04:34.610","Text":"The idea we did there was to change sine squared to 1 minus cosine squared."},{"Start":"04:34.610 ","End":"04:39.920","Text":"If we change sine squared to 1 minus cosine squared,"},{"Start":"04:39.920 ","End":"04:44.180","Text":"we got 1 minus cosine squared t over"},{"Start":"04:44.180 ","End":"04:51.765","Text":"cosine cubed t. Then we split it up into 2 parts,"},{"Start":"04:51.765 ","End":"05:01.720","Text":"and we got 1 over cosine cubed minus cosine squared over cosine cubed is just cosine."},{"Start":"05:01.720 ","End":"05:05.444","Text":"We broke it up into 2 bits."},{"Start":"05:05.444 ","End":"05:08.030","Text":"We finally ended up with,"},{"Start":"05:08.030 ","End":"05:11.090","Text":"and I\u0027m just quoting the previous exercise,"},{"Start":"05:11.090 ","End":"05:15.440","Text":"just copied the result from the previous exercise."},{"Start":"05:15.440 ","End":"05:21.709","Text":"What we said we\u0027re going to do there is just replace t by what\u0027s written here,"},{"Start":"05:21.709 ","End":"05:24.245","Text":"here, here, here, and here."},{"Start":"05:24.245 ","End":"05:27.950","Text":"We can combine the 4 with 1.5 and we get to and"},{"Start":"05:27.950 ","End":"05:33.905","Text":"whatever by substituting t plus c. Look at the previous exercise."},{"Start":"05:33.905 ","End":"05:36.420","Text":"Okay, we\u0027re done."}],"ID":6733},{"Watched":false,"Name":"Exercise 7","Duration":"6m 32s","ChapterTopicVideoID":6673,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Here we have to compute this integral."},{"Start":"00:02.250 ","End":"00:04.635","Text":"I just copied it over here."},{"Start":"00:04.635 ","End":"00:11.325","Text":"The first thing we try is letting t equals the square root of this thing."},{"Start":"00:11.325 ","End":"00:14.535","Text":"Substitution. I already tried it, it doesn\u0027t work."},{"Start":"00:14.535 ","End":"00:18.045","Text":"The next thing to try is a trigonometric substitution."},{"Start":"00:18.045 ","End":"00:21.450","Text":"But this doesn\u0027t look like 1 of the standard forms."},{"Start":"00:21.450 ","End":"00:27.165","Text":"The closest it looks like is the a squared minus x squared."},{"Start":"00:27.165 ","End":"00:30.990","Text":"But it\u0027s just not going to work because we have an x term."},{"Start":"00:30.990 ","End":"00:34.260","Text":"What I\u0027d like to try is to see if I can get it in the form of"},{"Start":"00:34.260 ","End":"00:39.435","Text":"a squared minus x plus something squared."},{"Start":"00:39.435 ","End":"00:41.250","Text":"Maybe that will work."},{"Start":"00:41.250 ","End":"00:44.295","Text":"We do this by completing the square."},{"Start":"00:44.295 ","End":"00:48.165","Text":"I\u0027ll just do this completing the square at the side."},{"Start":"00:48.165 ","End":"00:49.875","Text":"Since it\u0027s going to be minus,"},{"Start":"00:49.875 ","End":"00:53.370","Text":"what I want to do is complete the square for 6x plus x"},{"Start":"00:53.370 ","End":"00:57.120","Text":"squared because then it\u0027s going to be all minus."},{"Start":"00:57.120 ","End":"00:58.875","Text":"What I have is minus of this."},{"Start":"00:58.875 ","End":"01:04.245","Text":"Well, I\u0027ll reverse the order x squared plus 6x."},{"Start":"01:04.245 ","End":"01:08.670","Text":"The way it works is you want it to be something squared."},{"Start":"01:08.670 ","End":"01:12.360","Text":"X plus something squared is x squared plus twice that something."},{"Start":"01:12.360 ","End":"01:18.915","Text":"So it\u0027ll have to be x plus 3 squared and 3 squared is 9,"},{"Start":"01:18.915 ","End":"01:21.735","Text":"then it will be x plus 3 squared."},{"Start":"01:21.735 ","End":"01:24.300","Text":"But I can\u0027t just add the 9."},{"Start":"01:24.300 ","End":"01:28.155","Text":"I also have to correct by subtracting 9."},{"Start":"01:28.155 ","End":"01:32.400","Text":"Now it will be x plus 3 squared and just check it,"},{"Start":"01:32.400 ","End":"01:36.975","Text":"multiply it out, x squared plus twice x times 3 plus 3 squared."},{"Start":"01:36.975 ","End":"01:41.235","Text":"The usual thing to do is to take half of this and square it,"},{"Start":"01:41.235 ","End":"01:46.620","Text":"and that will give you the perfect square and then compensate minus 9."},{"Start":"01:46.620 ","End":"01:49.020","Text":"Now here, because it\u0027s in minus,"},{"Start":"01:49.020 ","End":"01:56.775","Text":"what I get is the integral of 9 minus x plus 3 squared because it\u0027s all reversed."},{"Start":"01:56.775 ","End":"02:06.600","Text":"The square root of 9 minus x plus 3 squared dx."},{"Start":"02:06.600 ","End":"02:13.365","Text":"Now this thing will be good for me if I take a equals 3,"},{"Start":"02:13.365 ","End":"02:16.289","Text":"that will be 3 squared minus,"},{"Start":"02:16.289 ","End":"02:19.875","Text":"well, not x squared, but something close."},{"Start":"02:19.875 ","End":"02:23.490","Text":"We need the equations for a squared minus x"},{"Start":"02:23.490 ","End":"02:26.850","Text":"squared and I\u0027ll go and copy paste them from another exercise,"},{"Start":"02:26.850 ","End":"02:28.350","Text":"we\u0027ve done this before."},{"Start":"02:28.350 ","End":"02:30.465","Text":"Here\u0027s the 3 equations,"},{"Start":"02:30.465 ","End":"02:33.330","Text":"but we have to modify them because instead of x,"},{"Start":"02:33.330 ","End":"02:34.785","Text":"we have x plus 3."},{"Start":"02:34.785 ","End":"02:42.465","Text":"What I\u0027m going to do is replace x with x plus 3 here and here, and here."},{"Start":"02:42.465 ","End":"02:48.000","Text":"But here we don\u0027t need any change because when it\u0027s x plus 3,"},{"Start":"02:48.000 ","End":"02:50.760","Text":"d of x plus 3 is the same."},{"Start":"02:50.760 ","End":"02:53.500","Text":"They have the same derivative which is 1."},{"Start":"02:53.870 ","End":"02:57.540","Text":"If I interpret these 3,"},{"Start":"02:57.540 ","End":"03:02.295","Text":"in our case, we get 1, 2, and 3."},{"Start":"03:02.295 ","End":"03:05.730","Text":"Let\u0027s see, x plus 3."},{"Start":"03:05.730 ","End":"03:08.865","Text":"You know what, maybe I\u0027ll change them over here. Hang on."},{"Start":"03:08.865 ","End":"03:12.870","Text":"X plus 3 equals, a is 3,"},{"Start":"03:12.870 ","End":"03:20.130","Text":"so it\u0027s 3 sine t. If we need the reverse substitution,"},{"Start":"03:20.130 ","End":"03:21.465","Text":"which we probably will,"},{"Start":"03:21.465 ","End":"03:30.570","Text":"then t is the arcsine of x plus 3 over 3 is 3."},{"Start":"03:30.570 ","End":"03:36.630","Text":"Number 2, square root of a squared is 9,"},{"Start":"03:36.630 ","End":"03:41.250","Text":"that\u0027s 3 squared minus x plus 3"},{"Start":"03:41.250 ","End":"03:49.095","Text":"squared is equal to 3 cosine t. The last 1,"},{"Start":"03:49.095 ","End":"03:55.240","Text":"dx is equal to 3 cosine tdt."},{"Start":"03:55.550 ","End":"04:00.840","Text":"Then I\u0027m going to plug them into here."},{"Start":"04:00.840 ","End":"04:02.835","Text":"This is equal to,"},{"Start":"04:02.835 ","End":"04:07.425","Text":"now we\u0027re going to use the substitutions, the integral."},{"Start":"04:07.425 ","End":"04:10.770","Text":"This bit we have already here."},{"Start":"04:10.770 ","End":"04:14.325","Text":"I\u0027m using formula 2 to say that this is 3,"},{"Start":"04:14.325 ","End":"04:23.590","Text":"cosine t and dx I\u0027ll get from formula 3 is 3 cosine tdt."},{"Start":"04:24.080 ","End":"04:26.490","Text":"Now I\u0027m going to tidy up a bit."},{"Start":"04:26.490 ","End":"04:30.890","Text":"This one I\u0027ll need, but the rest I\u0027ll erase. More space."},{"Start":"04:30.890 ","End":"04:35.960","Text":"Now I can continue and we get 3 times 3 is 9."},{"Start":"04:35.960 ","End":"04:38.015","Text":"I\u0027ll take it outside the integral."},{"Start":"04:38.015 ","End":"04:42.115","Text":"Cosine times cosine is cosine squared."},{"Start":"04:42.115 ","End":"04:48.420","Text":"Here we use a standard trick of trigonometrical identity for cosine squared."},{"Start":"04:48.420 ","End":"04:52.080","Text":"You must have a formula sheet somewhere with trigonometrical identities."},{"Start":"04:52.080 ","End":"04:57.600","Text":"In any event, this is equal to 1 plus cosine 2t."},{"Start":"04:57.600 ","End":"05:00.240","Text":"Usually it\u0027s with Alpha in the formula sheet,"},{"Start":"05:00.240 ","End":"05:04.140","Text":"but it could be t over 2."},{"Start":"05:04.140 ","End":"05:09.240","Text":"Now we can do this because each of these is immediate."},{"Start":"05:09.240 ","End":"05:12.165","Text":"Let\u0027s just write it as 9 over 2,"},{"Start":"05:12.165 ","End":"05:13.890","Text":"and then we\u0027ll write it as 2 bits."},{"Start":"05:13.890 ","End":"05:18.675","Text":"The first bit is 1 dt or just dt."},{"Start":"05:18.675 ","End":"05:25.170","Text":"The second bit is the integral of cosine 2t dt."},{"Start":"05:25.170 ","End":"05:29.715","Text":"This is equal to 9 over 2."},{"Start":"05:29.715 ","End":"05:37.650","Text":"Now the integral of 1 is just t. The integral of cosine 2t is sine 2t,"},{"Start":"05:37.650 ","End":"05:43.260","Text":"but not quite because it\u0027s 2t we need to divide by 2 and plus"},{"Start":"05:43.260 ","End":"05:50.320","Text":"C. That\u0027s not the end because we have to substitute from t back to arcsine."},{"Start":"05:50.630 ","End":"05:55.200","Text":"This will equal 9 over 2."},{"Start":"05:55.200 ","End":"06:00.900","Text":"T is arcsine of x plus 3 over 3."},{"Start":"06:00.900 ","End":"06:02.430","Text":"Just going to copy that,"},{"Start":"06:02.430 ","End":"06:11.205","Text":"arcsine x plus 3 over 3 plus 1/2 the sine of 2t,"},{"Start":"06:11.205 ","End":"06:18.480","Text":"the sine of 2 arcsine x plus 3 over"},{"Start":"06:18.480 ","End":"06:26.160","Text":"3 plus C. There are ways of simplifying this to get rid of the trigonometrical stuff."},{"Start":"06:26.160 ","End":"06:27.660","Text":"But this will still remain,"},{"Start":"06:27.660 ","End":"06:33.010","Text":"arcsine and basically this is okay as an answer and I\u0027m done."}],"ID":6734},{"Watched":false,"Name":"Exercise 8","Duration":"7m ","ChapterTopicVideoID":6674,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"Here we have to compute this integral which I\u0027ve rewritten over here."},{"Start":"00:04.755 ","End":"00:07.605","Text":"It\u0027s not immediately clear how to solve this,"},{"Start":"00:07.605 ","End":"00:09.570","Text":"but the fact that it\u0027s in the chapter on"},{"Start":"00:09.570 ","End":"00:14.700","Text":"trigonometric substitutions might lead us to try a trigonometric substitution,"},{"Start":"00:14.700 ","End":"00:18.300","Text":"but the trigonometric substitutions have a square root in them."},{"Start":"00:18.300 ","End":"00:19.845","Text":"If you ignore that,"},{"Start":"00:19.845 ","End":"00:29.650","Text":"then this is very close to the square root of a squared plus x squared with a being 2."},{"Start":"00:29.650 ","End":"00:33.765","Text":"However, here there\u0027s a square root and here there\u0027s not."},{"Start":"00:33.765 ","End":"00:38.300","Text":"What can we do? We can play with the exponents."},{"Start":"00:38.300 ","End":"00:42.800","Text":"I mean, the square root is just to the power of a 1/2."},{"Start":"00:42.800 ","End":"00:46.040","Text":"This is a squared plus x squared to the power of a 1/2,"},{"Start":"00:46.040 ","End":"00:51.210","Text":"and this is to the power of minus 2."},{"Start":"00:56.020 ","End":"00:58.540","Text":"I want to get minus 2,"},{"Start":"00:58.540 ","End":"01:01.640","Text":"so if I multiply it by minus 4,"},{"Start":"01:01.640 ","End":"01:06.350","Text":"if I take this a squared plus x squared to the power of"},{"Start":"01:06.350 ","End":"01:11.755","Text":"1/2 and I raise it to the power of minus 4,"},{"Start":"01:11.755 ","End":"01:19.730","Text":"I\u0027m working backwards, I get a squared plus x squared to the minus 2,"},{"Start":"01:19.730 ","End":"01:21.080","Text":"which is what I have here."},{"Start":"01:21.080 ","End":"01:26.640","Text":"I mean, this is just 4 plus x squared to the minus 2."},{"Start":"01:26.830 ","End":"01:30.620","Text":"I just did the minus 2 over a half and got"},{"Start":"01:30.620 ","End":"01:35.060","Text":"minus 4 or to said a half times what gives me minus 2."},{"Start":"01:35.060 ","End":"01:38.590","Text":"Now this is the square root, this bit."},{"Start":"01:38.590 ","End":"01:43.250","Text":"What I\u0027m saying is this is equal to the square root of"},{"Start":"01:43.250 ","End":"01:50.690","Text":"4 plus x squared to the power of minus 4 dx."},{"Start":"01:50.690 ","End":"01:53.240","Text":"If you think about it, this is to the power of a half."},{"Start":"01:53.240 ","End":"01:58.830","Text":"A half times minus 4 is minus 2 and that gives me this."},{"Start":"01:59.210 ","End":"02:03.260","Text":"I just copied and pasted from a previous exercise"},{"Start":"02:03.260 ","End":"02:06.695","Text":"the 3 equations for a squared plus x squared."},{"Start":"02:06.695 ","End":"02:08.160","Text":"Since a equals 2,"},{"Start":"02:08.160 ","End":"02:10.985","Text":"I just want to substitute everywhere I see a,"},{"Start":"02:10.985 ","End":"02:12.260","Text":"I want to substitute 2,"},{"Start":"02:12.260 ","End":"02:16.090","Text":"so I get 1, 2, and 3."},{"Start":"02:16.090 ","End":"02:25.025","Text":"The first one with a equals 2 gives me x equals 2 tangent t and the reverse substitution,"},{"Start":"02:25.025 ","End":"02:30.670","Text":"t equals arctangent of x over 2."},{"Start":"02:30.670 ","End":"02:35.515","Text":"The 2nd one with a equals 2 gives me that the square root of"},{"Start":"02:35.515 ","End":"02:41.215","Text":"2 squared is 4 plus x squared is equal to 2"},{"Start":"02:41.215 ","End":"02:46.840","Text":"over cosine t. The last one tells me that dx"},{"Start":"02:46.840 ","End":"02:53.630","Text":"is equal to 2 over cosine squared t dt."},{"Start":"02:53.630 ","End":"02:57.700","Text":"Now what I want to do is use these to simplify this."},{"Start":"02:57.700 ","End":"02:59.110","Text":"This is going to equal,"},{"Start":"02:59.110 ","End":"03:00.250","Text":"now lets see what I can use."},{"Start":"03:00.250 ","End":"03:08.845","Text":"The square root of 4 plus x squared is 2 over cosine t. What I get is the integral"},{"Start":"03:08.845 ","End":"03:19.190","Text":"of 2 over cosine t to the power of minus 4."},{"Start":"03:19.190 ","End":"03:22.310","Text":"Dx, I can get from here,"},{"Start":"03:22.310 ","End":"03:30.180","Text":"and this is 2 over cosine t dt."},{"Start":"03:30.180 ","End":"03:32.930","Text":"Let me erase the extra stuff,"},{"Start":"03:32.930 ","End":"03:38.495","Text":"which is all of this except for this one I\u0027ll keep over here. That\u0027s better."},{"Start":"03:38.495 ","End":"03:43.350","Text":"Continuing, now to the power of minus 4."},{"Start":"03:43.350 ","End":"03:49.150","Text":"Again, I have 2 over cosine t so it just gives me this to the power of minus 3."},{"Start":"03:49.150 ","End":"03:56.955","Text":"It\u0027s 2 over cosine t to the minus 3 dt."},{"Start":"03:56.955 ","End":"04:04.680","Text":"This equals 2 to the minus 3 is 1/8,"},{"Start":"04:04.680 ","End":"04:07.034","Text":"which I can pull out in front,"},{"Start":"04:07.034 ","End":"04:11.640","Text":"2 cubed is 8, so 2 to the minus 3 is 8 times the integral."},{"Start":"04:11.640 ","End":"04:16.675","Text":"Now, 1 over cosine t to the minus 3,"},{"Start":"04:16.675 ","End":"04:20.255","Text":"because it\u0027s 1 over x to the power of minus 1."},{"Start":"04:20.255 ","End":"04:27.605","Text":"If you think about it, we just get cosine cubed t with a positive exponent dt."},{"Start":"04:27.605 ","End":"04:29.930","Text":"Now we just have to remember"},{"Start":"04:29.930 ","End":"04:35.870","Text":"our trigonometric integrals and the standard bag of tricks and identities,"},{"Start":"04:35.870 ","End":"04:39.920","Text":"in this case at least it seems to me fairly clear that"},{"Start":"04:39.920 ","End":"04:44.710","Text":"we want to take cosine squared and write it as 1 minus sine squared."},{"Start":"04:44.710 ","End":"04:48.665","Text":"This is cosine squared times cosine."},{"Start":"04:48.665 ","End":"04:49.925","Text":"That\u0027s the cosine cubed,"},{"Start":"04:49.925 ","End":"04:51.695","Text":"I just saved myself a step."},{"Start":"04:51.695 ","End":"04:55.865","Text":"The cosine squared is 1 minus sine squared."},{"Start":"04:55.865 ","End":"05:04.160","Text":"I get 1 minus sine squared t times cosine t dt."},{"Start":"05:04.160 ","End":"05:06.290","Text":"Let\u0027s do this one by a substitution,"},{"Start":"05:06.290 ","End":"05:09.200","Text":"since the derivative of sine t is cosine t,"},{"Start":"05:09.200 ","End":"05:12.050","Text":"I think I\u0027d like to substitute sine t for something,"},{"Start":"05:12.050 ","End":"05:13.325","Text":"let\u0027s call it say,"},{"Start":"05:13.325 ","End":"05:20.370","Text":"u is equal to sine t and then du will equal"},{"Start":"05:20.370 ","End":"05:27.150","Text":"cosine t dt and that will give us"},{"Start":"05:27.150 ","End":"05:34.790","Text":"1/8 the integral of 1 minus sine t is u,"},{"Start":"05:34.790 ","End":"05:37.280","Text":"so that\u0027s u squared,"},{"Start":"05:37.280 ","End":"05:41.960","Text":"and cosine t dt is du."},{"Start":"05:41.960 ","End":"05:49.470","Text":"This gives us 1/8 of u minus u cubed over"},{"Start":"05:49.470 ","End":"05:58.055","Text":"3 plus c. Now we substitute back that u is sine t,"},{"Start":"05:58.055 ","End":"06:04.160","Text":"so it\u0027s 1/8 of sine t minus 1/3, I\u0027ll just write it this way,"},{"Start":"06:04.160 ","End":"06:09.950","Text":"sine cubed t. Now we have to make another substitution"},{"Start":"06:09.950 ","End":"06:16.790","Text":"back using this bit here that t is arctangent of x over 2."},{"Start":"06:16.790 ","End":"06:20.970","Text":"What we get is 1/8."},{"Start":"06:20.970 ","End":"06:26.190","Text":"It\u0027s sine of arctangent of x"},{"Start":"06:26.190 ","End":"06:33.225","Text":"over 2 minus 1/3 sine of arctangent."},{"Start":"06:33.225 ","End":"06:36.420","Text":"I just forgot the sine here."},{"Start":"06:36.420 ","End":"06:39.960","Text":"Arc tangent of x over 2."},{"Start":"06:39.960 ","End":"06:46.785","Text":"This is sine cubed of arc tangent x over 2 plus the constant."},{"Start":"06:46.785 ","End":"06:52.400","Text":"This could be simplified using trigonometry of right triangles,"},{"Start":"06:52.400 ","End":"06:54.125","Text":"but I\u0027m not going to do that."},{"Start":"06:54.125 ","End":"06:56.645","Text":"I\u0027ll leave this as the answer and we\u0027re done."},{"Start":"06:56.645 ","End":"07:00.630","Text":"Though I forgot the brackets, not terribly important."}],"ID":6735},{"Watched":false,"Name":"Exercise 9","Duration":"5m 12s","ChapterTopicVideoID":4398,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Here, we have to compute the following integral,"},{"Start":"00:02.640 ","End":"00:04.335","Text":"which I\u0027ve copied over here."},{"Start":"00:04.335 ","End":"00:08.445","Text":"The usual thing to do is to try a substitution t equals this thing."},{"Start":"00:08.445 ","End":"00:10.170","Text":"I\u0027ve tried it, it doesn\u0027t work."},{"Start":"00:10.170 ","End":"00:14.400","Text":"Next thing to try is trigonometrical substitution."},{"Start":"00:14.400 ","End":"00:17.040","Text":"By the look of it, it\u0027s going to be 1 of those complete"},{"Start":"00:17.040 ","End":"00:19.545","Text":"the square problems which you should be familiar with."},{"Start":"00:19.545 ","End":"00:21.390","Text":"I\u0027m going to do it a bit quicker."},{"Start":"00:21.390 ","End":"00:24.060","Text":"I\u0027m going to complete the square at the side here."},{"Start":"00:24.060 ","End":"00:29.565","Text":"Try to get it to be x squared plus a squared or x squared minus a squared, that thing."},{"Start":"00:29.565 ","End":"00:34.800","Text":"I have that x squared plus 2x plus 5."},{"Start":"00:34.800 ","End":"00:37.030","Text":"I want to complete the square."},{"Start":"00:37.030 ","End":"00:39.200","Text":"What we do is, x plus, now,"},{"Start":"00:39.200 ","End":"00:42.960","Text":"1/2 of this coefficient is 1 squared,"},{"Start":"00:42.960 ","End":"00:45.350","Text":"and then we can multiply it out in our heads,"},{"Start":"00:45.350 ","End":"00:48.605","Text":"x squared plus 2x plus 1."},{"Start":"00:48.605 ","End":"00:51.575","Text":"I have from here plus 1, I need plus 5,"},{"Start":"00:51.575 ","End":"00:55.869","Text":"so I add plus 4 and this 4 is 2 squared."},{"Start":"00:55.869 ","End":"01:00.170","Text":"Indeed we do get an x squared plus a squared."},{"Start":"01:00.170 ","End":"01:10.580","Text":"What we have is the integral of 1 over x plus 1 squared plus 4,"},{"Start":"01:10.580 ","End":"01:12.570","Text":"which is 2 squared."},{"Start":"01:12.570 ","End":"01:14.645","Text":"I could leave it as 4, it doesn\u0027t matter,"},{"Start":"01:14.645 ","End":"01:16.385","Text":"to the power of 1 and 1/2."},{"Start":"01:16.385 ","End":"01:18.545","Text":"I don\u0027t like decimals and exponents."},{"Start":"01:18.545 ","End":"01:21.805","Text":"Make it 3 over 2 dx."},{"Start":"01:21.805 ","End":"01:29.780","Text":"We have the basic case of x squared plus a squared."},{"Start":"01:29.780 ","End":"01:33.095","Text":"In this case, with a equals 2,"},{"Start":"01:33.095 ","End":"01:37.415","Text":"only it\u0027s not x, it\u0027s x plus 1. Make a note to that."},{"Start":"01:37.415 ","End":"01:40.340","Text":"Instead of x, we have to have x plus 1."},{"Start":"01:40.340 ","End":"01:44.194","Text":"Let me write the 3 equations for this format."},{"Start":"01:44.194 ","End":"01:49.505","Text":"Sorry, I meant to say the square root of x squared plus a squared."},{"Start":"01:49.505 ","End":"01:53.880","Text":"Not too late. Here are the 3 equations."},{"Start":"01:53.880 ","End":"01:55.730","Text":"What we have to do is interpret them."},{"Start":"01:55.730 ","End":"01:57.410","Text":"In our case, we have to do 2 things."},{"Start":"01:57.410 ","End":"01:59.570","Text":"You have to replace a by 2,"},{"Start":"01:59.570 ","End":"02:02.480","Text":"and we have to replace x by x plus 1. Let\u0027s see."},{"Start":"02:02.480 ","End":"02:06.735","Text":"Here, here, and here,"},{"Start":"02:06.735 ","End":"02:11.465","Text":"but here we don\u0027t need 2 because d of x plus 1 is the same as d of x."},{"Start":"02:11.465 ","End":"02:15.140","Text":"Internal derivatives 1. To save time,"},{"Start":"02:15.140 ","End":"02:16.430","Text":"I did it for you."},{"Start":"02:16.430 ","End":"02:20.810","Text":"I replaced everywhere a by 2 and wherever we had x,"},{"Start":"02:20.810 ","End":"02:22.550","Text":"I replaced it by x plus 1."},{"Start":"02:22.550 ","End":"02:26.895","Text":"That\u0027s here, here, and here."},{"Start":"02:26.895 ","End":"02:30.479","Text":"Let\u0027s see what it means in our case."},{"Start":"02:30.479 ","End":"02:33.650","Text":"There\u0027s 1 more thing slightly confusing."},{"Start":"02:33.650 ","End":"02:35.615","Text":"I wasn\u0027t completely careful with it."},{"Start":"02:35.615 ","End":"02:38.690","Text":"The standard formula is not x squared plus a squared,"},{"Start":"02:38.690 ","End":"02:40.835","Text":"but a squared plus x squared."},{"Start":"02:40.835 ","End":"02:43.729","Text":"Let me replace that here."},{"Start":"02:43.729 ","End":"02:48.185","Text":"Let me also change the order of the addition here."},{"Start":"02:48.185 ","End":"02:51.995","Text":"Then write 2 squared as 4 because it looks like the 4 here."},{"Start":"02:51.995 ","End":"02:55.465","Text":"I\u0027ll write this as 4 plus x plus 1 squared."},{"Start":"02:55.465 ","End":"02:59.510","Text":"Now we\u0027re all straight with the a squared plus x squared."},{"Start":"02:59.510 ","End":"03:01.610","Text":"The question is how to substitute."},{"Start":"03:01.610 ","End":"03:04.850","Text":"I don\u0027t have a square root apparently here,"},{"Start":"03:04.850 ","End":"03:09.060","Text":"but I do have something to the power of 3 over 2."},{"Start":"03:09.310 ","End":"03:11.825","Text":"I\u0027ll just do this at the side."},{"Start":"03:11.825 ","End":"03:17.330","Text":"If I take a^3 over 2,"},{"Start":"03:17.330 ","End":"03:22.550","Text":"this is just a^1/2^3,"},{"Start":"03:22.550 ","End":"03:28.135","Text":"and a^1/2 is the square root of a^3."},{"Start":"03:28.135 ","End":"03:33.920","Text":"What I really have here is the square root of this thing to the power of 3."},{"Start":"03:34.170 ","End":"03:38.600","Text":"I can write this denominator"},{"Start":"03:38.600 ","End":"03:43.720","Text":"as the square root of 4 plus x plus 1 squared to the power of 3,"},{"Start":"03:43.720 ","End":"03:52.620","Text":"which means 2 over cosine t to the power of 3, using rule 2."},{"Start":"03:52.620 ","End":"03:56.200","Text":"Now I\u0027m going to use rule 3 to replace dx,"},{"Start":"03:56.200 ","End":"03:59.395","Text":"and dx is 2 over cosine squared dt."},{"Start":"03:59.395 ","End":"04:05.045","Text":"I\u0027ll put the dt at the side and the 2 over cosine squared t here,"},{"Start":"04:05.045 ","End":"04:13.580","Text":"cosine squared t. Now I\u0027ve got my integral in terms of t and I need to simplify,"},{"Start":"04:13.580 ","End":"04:16.120","Text":"but all this stuff is in the way."},{"Start":"04:16.120 ","End":"04:18.135","Text":"Now, what we have here,"},{"Start":"04:18.135 ","End":"04:20.915","Text":"and I want to simplify this, is,"},{"Start":"04:20.915 ","End":"04:23.390","Text":"let\u0027s see, as far as the numbers go,"},{"Start":"04:23.390 ","End":"04:25.820","Text":"it\u0027s 2 over 2 cubed."},{"Start":"04:25.820 ","End":"04:28.155","Text":"If you figure that out, it\u0027s 1/4,"},{"Start":"04:28.155 ","End":"04:30.720","Text":"which I can bring to the front."},{"Start":"04:30.720 ","End":"04:36.965","Text":"I have cosine squared goes denominator and cosine cubed goes to the numerator."},{"Start":"04:36.965 ","End":"04:41.060","Text":"I end up with just cosine t dt, how very nice."},{"Start":"04:41.060 ","End":"04:44.644","Text":"Now, the integral of cosine is an immediate integral,"},{"Start":"04:44.644 ","End":"04:48.260","Text":"is sine t plus a constant."},{"Start":"04:48.260 ","End":"04:52.970","Text":"The only thing we have left to do is to replace t from this."},{"Start":"04:52.970 ","End":"04:58.430","Text":"The answer is, 1/4 sine of"},{"Start":"04:58.430 ","End":"05:05.320","Text":"arctangent of x plus 1 over 2 plus a constant."},{"Start":"05:05.320 ","End":"05:09.425","Text":"It is possible to simplify this and get rid of the trigonometric stuff,"},{"Start":"05:09.425 ","End":"05:10.955","Text":"but it\u0027s not necessary."},{"Start":"05:10.955 ","End":"05:13.230","Text":"We\u0027re going to end here."}],"ID":4407},{"Watched":false,"Name":"Exercise 10","Duration":"7m 12s","ChapterTopicVideoID":6675,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.685","Text":"Here we have to compute this integral,"},{"Start":"00:02.685 ","End":"00:04.920","Text":"which I\u0027ve copied over here."},{"Start":"00:04.920 ","End":"00:07.320","Text":"The first thing you would normally try would be"},{"Start":"00:07.320 ","End":"00:10.875","Text":"a substitution for t equals the square root of this thing."},{"Start":"00:10.875 ","End":"00:12.570","Text":"But I\u0027ve tried it and it doesn\u0027t work."},{"Start":"00:12.570 ","End":"00:15.810","Text":"The next thing we try when you have square roots"},{"Start":"00:15.810 ","End":"00:19.420","Text":"and squares is trigonometric substitution."},{"Start":"00:19.420 ","End":"00:21.170","Text":"There are 3 basic types,"},{"Start":"00:21.170 ","End":"00:29.030","Text":"and this most closely looks like the x squared minus a squared, but not quite."},{"Start":"00:29.030 ","End":"00:32.285","Text":"Let me show you how we modify this."},{"Start":"00:32.285 ","End":"00:37.340","Text":"We have the square root of 25x squared minus 4."},{"Start":"00:37.340 ","End":"00:39.470","Text":"We want it to look like this."},{"Start":"00:39.470 ","End":"00:44.095","Text":"What I do is I write this as the square root of,"},{"Start":"00:44.095 ","End":"00:53.755","Text":"I take 25 outside the brackets and I\u0027m left with x squared minus 4/25."},{"Start":"00:53.755 ","End":"00:56.480","Text":"Then the square root of a product is the product of"},{"Start":"00:56.480 ","End":"00:59.525","Text":"the square root so the square root of 25 is 5."},{"Start":"00:59.525 ","End":"01:06.725","Text":"I\u0027m left with the square root of x squared minus 4/25,"},{"Start":"01:06.725 ","End":"01:10.310","Text":"which I can also write as the square root of x squared"},{"Start":"01:10.310 ","End":"01:17.305","Text":"minus 2/5 squared because the square root of 4 is 2 and the square root of 25 is 5."},{"Start":"01:17.305 ","End":"01:19.930","Text":"This would be more convenient."},{"Start":"01:19.930 ","End":"01:27.440","Text":"I\u0027m going to rewrite this integral using these side calculations in the following form."},{"Start":"01:27.440 ","End":"01:30.005","Text":"This will be the integral of,"},{"Start":"01:30.005 ","End":"01:37.070","Text":"now the 5 can come all the way out here and I have the square root of x"},{"Start":"01:37.070 ","End":"01:41.150","Text":"squared minus 2/5"},{"Start":"01:41.150 ","End":"01:47.740","Text":"squared over x dx."},{"Start":"01:47.740 ","End":"01:54.080","Text":"The basic substitution we want to use is the square root of"},{"Start":"01:54.080 ","End":"02:01.015","Text":"x squared minus a squared with a equaling 2/5."},{"Start":"02:01.015 ","End":"02:03.680","Text":"We\u0027ve done this thing before."},{"Start":"02:03.680 ","End":"02:07.715","Text":"Let me go and copy paste from another exercise."},{"Start":"02:07.715 ","End":"02:11.645","Text":"Here they are, the 3 equations for the substitution."},{"Start":"02:11.645 ","End":"02:13.310","Text":"Let\u0027s interpret them."},{"Start":"02:13.310 ","End":"02:16.910","Text":"In our case we have 1, 2, and 3."},{"Start":"02:16.910 ","End":"02:23.525","Text":"What we have to do is just replace a by 2/5 in this standard so what do we get?"},{"Start":"02:23.525 ","End":"02:31.640","Text":"x is equal to 2 over 5 cosine t."},{"Start":"02:31.640 ","End":"02:37.575","Text":"The reverse substitution is t equals arccosine"},{"Start":"02:37.575 ","End":"02:39.720","Text":"a/x which"},{"Start":"02:39.720 ","End":"02:48.090","Text":"is 2/5x."},{"Start":"02:48.090 ","End":"02:54.110","Text":"Number 2, the square root of x squared,"},{"Start":"02:54.110 ","End":"02:58.250","Text":"I can write it as 4/25 or 2/5 squared,"},{"Start":"02:58.250 ","End":"03:07.640","Text":"doesn\u0027t matter and that equals a 2/5 tangent of t and the last one,"},{"Start":"03:07.640 ","End":"03:11.030","Text":"dx is equal to, a is 2/5,"},{"Start":"03:11.030 ","End":"03:20.620","Text":"so it\u0027s 2 sine t over 5 cosine squared t dt."},{"Start":"03:20.930 ","End":"03:26.250","Text":"With these 3, I want to simplify this integral."},{"Start":"03:26.250 ","End":"03:30.375","Text":"This is equal to 5 times the integral of,"},{"Start":"03:30.375 ","End":"03:35.030","Text":"now this numerator is exactly number 2."},{"Start":"03:35.030 ","End":"03:45.210","Text":"In the numerator I\u0027m going to put 2/5 tangent of t. In the denominator I have x and that\u0027s"},{"Start":"03:45.210 ","End":"03:51.150","Text":"exactly number 1 so it\u0027s 2 over 5 cosine t or if I"},{"Start":"03:51.150 ","End":"03:58.310","Text":"like 2/5 1 over cosine t. The final thing is the dx, that\u0027s number 3."},{"Start":"03:58.310 ","End":"04:02.360","Text":"Number 3 is 2/5,"},{"Start":"04:02.360 ","End":"04:04.820","Text":"I like to write the 2/5 separately."},{"Start":"04:04.820 ","End":"04:13.075","Text":"Then sin t over cosine squared t dt."},{"Start":"04:13.075 ","End":"04:18.230","Text":"Next, I want to clear this so I can have room to continue and I won\u0027t need any of"},{"Start":"04:18.230 ","End":"04:24.315","Text":"this except this which I\u0027ll put over here. Here we are."},{"Start":"04:24.315 ","End":"04:28.975","Text":"All we have to do is simplify this a bit as I also scroll up."},{"Start":"04:28.975 ","End":"04:32.650","Text":"What do we get? Let\u0027s take care of the constants first."},{"Start":"04:32.650 ","End":"04:35.155","Text":"Now, some things cancel."},{"Start":"04:35.155 ","End":"04:39.315","Text":"This 2/5 will cancel with this 2/5,"},{"Start":"04:39.315 ","End":"04:42.899","Text":"and this 5 will cancel with this 5,"},{"Start":"04:42.899 ","End":"04:49.240","Text":"and the 2 will come out front so I\u0027ll get twice the integral."},{"Start":"04:49.240 ","End":"04:51.265","Text":"Now let\u0027s see."},{"Start":"04:51.265 ","End":"04:56.945","Text":"What we have is this 1 over cosine t together with"},{"Start":"04:56.945 ","End":"05:04.755","Text":"this cosine t will give me cosine t in the denominator."},{"Start":"05:04.755 ","End":"05:13.795","Text":"In the numerator, I\u0027ll get tangent t times sine t dt."},{"Start":"05:13.795 ","End":"05:19.990","Text":"But we can go a little bit further because tangent is sine over cosine."},{"Start":"05:19.990 ","End":"05:28.580","Text":"Sine over cosine means I\u0027ll get the integral of sine squared over cosine squared."},{"Start":"05:28.580 ","End":"05:36.565","Text":"I have sine squared t over cosine squared t dt."},{"Start":"05:36.565 ","End":"05:40.805","Text":"This is a trigonometric integral using identities."},{"Start":"05:40.805 ","End":"05:46.285","Text":"The first thing to do would be to write the sine squared as 1 minus cosine squared."},{"Start":"05:46.285 ","End":"05:54.040","Text":"I\u0027ve got 1 minus cosine squared t over cosine squared t dt."},{"Start":"05:54.040 ","End":"05:57.770","Text":"Now, because of this minus here,"},{"Start":"05:57.770 ","End":"06:01.555","Text":"I can separate it into 2 integrals."},{"Start":"06:01.555 ","End":"06:04.710","Text":"What I get is twice,"},{"Start":"06:04.710 ","End":"06:11.540","Text":"the first one is the integral of 1 over cosine squared t dt,"},{"Start":"06:11.540 ","End":"06:13.625","Text":"which is actually an immediate integral,"},{"Start":"06:13.625 ","End":"06:18.110","Text":"and the other one is cosine squared over cosine squared,"},{"Start":"06:18.110 ","End":"06:20.945","Text":"and that\u0027s just 1 dt."},{"Start":"06:20.945 ","End":"06:23.675","Text":"Now this is equal to twice."},{"Start":"06:23.675 ","End":"06:25.250","Text":"This is an immediate integral,"},{"Start":"06:25.250 ","End":"06:29.930","Text":"1 over cosine squared is tangent and the integral of 1,"},{"Start":"06:29.930 ","End":"06:33.525","Text":"of course, is just t plus a constant."},{"Start":"06:33.525 ","End":"06:39.800","Text":"The last thing we have to do is go back by replacing t with what it is in terms of x."},{"Start":"06:39.800 ","End":"06:44.880","Text":"Finally, we get that this is equal to twice,"},{"Start":"06:45.070 ","End":"06:49.035","Text":"I forgot the tangent there,"},{"Start":"06:49.035 ","End":"06:58.560","Text":"of arccosine of 2/5x minus just t,"},{"Start":"06:58.560 ","End":"07:07.575","Text":"which is arccosine of 2/5x and all this plus a constant."},{"Start":"07:07.575 ","End":"07:11.105","Text":"It is possible to do some simplification on this term."},{"Start":"07:11.105 ","End":"07:13.800","Text":"We\u0027re not going to bother, we\u0027re done."}],"ID":6736},{"Watched":false,"Name":"Exercise 11","Duration":"11m 13s","ChapterTopicVideoID":4400,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.700","Text":"We have to compute the following definite integral,"},{"Start":"00:03.700 ","End":"00:07.220","Text":"which is slightly different than an indefinite integral."},{"Start":"00:07.310 ","End":"00:10.810","Text":"If you try irregular substitution,"},{"Start":"00:10.810 ","End":"00:12.010","Text":"you\u0027ll see it doesn\u0027t work."},{"Start":"00:12.010 ","End":"00:15.175","Text":"What we need is a trigonometric substitution."},{"Start":"00:15.175 ","End":"00:16.840","Text":"There are 3 basic types,"},{"Start":"00:16.840 ","End":"00:24.265","Text":"and this one most closely resembles the x squared minus a squared format."},{"Start":"00:24.265 ","End":"00:28.390","Text":"Now there is a formula for x squared minus a squared,"},{"Start":"00:28.390 ","End":"00:30.715","Text":"but it isn\u0027t quite in that form,"},{"Start":"00:30.715 ","End":"00:33.785","Text":"so we\u0027re going to have to do a little bit of algebra."},{"Start":"00:33.785 ","End":"00:42.730","Text":"Just notice that the square root of 25x squared minus 4,"},{"Start":"00:42.730 ","End":"00:46.370","Text":"we can rewrite it as the square root,"},{"Start":"00:46.370 ","End":"00:49.040","Text":"now I want to take 25 outside the brackets,"},{"Start":"00:49.040 ","End":"00:50.840","Text":"so I just have x squared,"},{"Start":"00:50.840 ","End":"00:55.105","Text":"and this would be minus 4 over 25."},{"Start":"00:55.105 ","End":"00:58.655","Text":"The square root of the product is the product of the square roots."},{"Start":"00:58.655 ","End":"01:00.904","Text":"The square root of 25 is 5,"},{"Start":"01:00.904 ","End":"01:07.505","Text":"and I\u0027m left with the square root of x squared minus 4 over 25."},{"Start":"01:07.505 ","End":"01:11.750","Text":"But this is just 2/5 squared."},{"Start":"01:11.750 ","End":"01:17.570","Text":"This now looks like this with a equals 2/5."},{"Start":"01:17.570 ","End":"01:26.195","Text":"I can rewrite this integral as the integral of,"},{"Start":"01:26.195 ","End":"01:32.400","Text":"write limits first from 2/5 to 4/5."},{"Start":"01:32.400 ","End":"01:34.370","Text":"This thing will equal this,"},{"Start":"01:34.370 ","End":"01:38.000","Text":"but I can take the 5 in front of the integral,"},{"Start":"01:38.000 ","End":"01:43.970","Text":"so what I\u0027m left with is the square root of x squared"},{"Start":"01:43.970 ","End":"01:52.440","Text":"minus 2/5 squared over x dx."},{"Start":"01:52.440 ","End":"01:57.335","Text":"Now I\u0027m going to use the substitution for x squared minus a squared."},{"Start":"01:57.335 ","End":"02:02.930","Text":"In our case, it\u0027s x squared minus a squared with a equals 2/5."},{"Start":"02:02.930 ","End":"02:07.730","Text":"There are standard equations for this trigonometric substitution,"},{"Start":"02:07.730 ","End":"02:10.870","Text":"and I\u0027ll copy-paste them from another exercise."},{"Start":"02:10.870 ","End":"02:13.500","Text":"Here are the equations,"},{"Start":"02:13.500 ","End":"02:18.770","Text":"but the ones we used for the indefinite"},{"Start":"02:18.770 ","End":"02:24.930","Text":"integral are not quite the same as the ones for the definite integral."},{"Start":"02:24.930 ","End":"02:31.580","Text":"Actually, the difference is that we don\u0027t have just the tangent of t here,"},{"Start":"02:31.580 ","End":"02:34.450","Text":"we actually have the absolute value."},{"Start":"02:34.450 ","End":"02:37.070","Text":"I\u0027m emphasizing this in a different color."},{"Start":"02:37.070 ","End":"02:41.030","Text":"But we can ignore this absolute value when we do indefinite integrals."},{"Start":"02:41.030 ","End":"02:43.685","Text":"We can\u0027t when we\u0027re doing definite integrals,"},{"Start":"02:43.685 ","End":"02:46.260","Text":"so we have to find out,"},{"Start":"02:47.720 ","End":"02:50.965","Text":"after we\u0027ve substituted t,"},{"Start":"02:50.965 ","End":"02:52.870","Text":"where we\u0027re positive and where we\u0027re negative,"},{"Start":"02:52.870 ","End":"02:54.140","Text":"it\u0027s something to watch out for."},{"Start":"02:54.140 ","End":"02:55.714","Text":"But see this difference,"},{"Start":"02:55.714 ","End":"03:00.140","Text":"that we use this in the case of definite integrals."},{"Start":"03:00.140 ","End":"03:08.345","Text":"Meanwhile, let\u0027s just interpret these 3 equations in terms of our case where a is 2/5."},{"Start":"03:08.345 ","End":"03:15.260","Text":"We get that x is equal to 2/5 over"},{"Start":"03:15.260 ","End":"03:23.955","Text":"cosine t. Sorry, it\u0027s 2/5."},{"Start":"03:23.955 ","End":"03:28.965","Text":"I\u0027m sorry. Here is 2 over 5 cosine t, sorry."},{"Start":"03:28.965 ","End":"03:38.870","Text":"T equals arccosine of a over x is 2 over 5x when a is 2/5."},{"Start":"03:38.870 ","End":"03:48.465","Text":"Number 2 is that the square root of x squared minus 2/5 squared"},{"Start":"03:48.465 ","End":"03:51.025","Text":"is equal to 2/5"},{"Start":"03:51.025 ","End":"03:54.035","Text":"times absolute value of"},{"Start":"03:54.035 ","End":"03:58.850","Text":"tangent t. Then we\u0027ll see what to do with the absolute value when we get to it."},{"Start":"03:58.850 ","End":"04:03.845","Text":"The last one says that dx is a,"},{"Start":"04:03.845 ","End":"04:08.690","Text":"which is 2/5, sine"},{"Start":"04:08.690 ","End":"04:15.195","Text":"t over cosine squared t, dt."},{"Start":"04:15.195 ","End":"04:20.180","Text":"I didn\u0027t leave enough room."},{"Start":"04:20.180 ","End":"04:24.920","Text":"I think what I\u0027ll do is scroll down first,"},{"Start":"04:24.920 ","End":"04:29.885","Text":"and then I\u0027ll just move these down a bit."},{"Start":"04:29.885 ","End":"04:32.510","Text":"That\u0027s better, more room."},{"Start":"04:32.510 ","End":"04:37.760","Text":"Continuing, this is equal to 5 times the integral."},{"Start":"04:37.760 ","End":"04:40.175","Text":"Now I can\u0027t write the 2/5 and the 4/5,"},{"Start":"04:40.175 ","End":"04:42.980","Text":"so I\u0027ll have to convert from x to t using this."},{"Start":"04:42.980 ","End":"04:47.265","Text":"But meanwhile, let\u0027s do the function and the dx."},{"Start":"04:47.265 ","End":"04:50.100","Text":"The square root of x squared minus 2/5,"},{"Start":"04:50.100 ","End":"04:53.830","Text":"that\u0027s 2/5 of tangent t,"},{"Start":"04:53.930 ","End":"04:56.330","Text":"but absolute value of"},{"Start":"04:56.330 ","End":"05:06.035","Text":"tangent t. X I get from here is 2 over 5 cosine t,"},{"Start":"05:06.035 ","End":"05:08.855","Text":"or if I like 2/5,"},{"Start":"05:08.855 ","End":"05:18.270","Text":"1 over cosine t. Dx is"},{"Start":"05:18.270 ","End":"05:23.610","Text":"2 sine t over"},{"Start":"05:23.610 ","End":"05:29.215","Text":"5 cosine squared t, dt."},{"Start":"05:29.215 ","End":"05:32.660","Text":"I\u0027ll substitute these in a moment,"},{"Start":"05:32.660 ","End":"05:36.090","Text":"let\u0027s just simplify first."},{"Start":"05:36.880 ","End":"05:39.290","Text":"These are getting in the way."},{"Start":"05:39.290 ","End":"05:41.030","Text":"I just need to keep this one."},{"Start":"05:41.030 ","End":"05:43.745","Text":"I\u0027ll put it over there and delete the rest."},{"Start":"05:43.745 ","End":"05:48.410","Text":"You know what, I think I\u0027ll compute the limits of the integration first."},{"Start":"05:48.410 ","End":"05:50.915","Text":"We have 2 cases;"},{"Start":"05:50.915 ","End":"05:56.070","Text":"we have to see that when x equals 2/5,"},{"Start":"05:56.070 ","End":"05:58.380","Text":"what does t equal,"},{"Start":"05:58.380 ","End":"06:01.815","Text":"and when x equals 4/5,"},{"Start":"06:01.815 ","End":"06:04.575","Text":"what does t equal?"},{"Start":"06:04.575 ","End":"06:07.440","Text":"I\u0027m using this formula down here."},{"Start":"06:07.440 ","End":"06:10.320","Text":"Here, t equals arccosine,"},{"Start":"06:10.320 ","End":"06:17.610","Text":"cosine of 2 over 5x."},{"Start":"06:17.610 ","End":"06:23.515","Text":"If I put x equals 2/5 here."},{"Start":"06:23.515 ","End":"06:31.510","Text":"I\u0027ll get 2 over 5 times 2/5."},{"Start":"06:33.560 ","End":"06:37.220","Text":"If you do the math, this is just equal to 1."},{"Start":"06:37.220 ","End":"06:38.720","Text":"Because 5 times 2/5 is 2."},{"Start":"06:38.720 ","End":"06:41.915","Text":"2 over 2 is 1. I\u0027ll just write 1 here."},{"Start":"06:41.915 ","End":"06:44.915","Text":"Now cosine of 1 is 0."},{"Start":"06:44.915 ","End":"06:50.030","Text":"You can see it by saying that cosine of 0 is 1."},{"Start":"06:50.030 ","End":"06:51.770","Text":"So our cosine of 1 is 0."},{"Start":"06:51.770 ","End":"06:54.260","Text":"You do it on your calculator."},{"Start":"06:54.260 ","End":"07:00.690","Text":"The other one, I\u0027m going to get if I put x equals 4/5."},{"Start":"07:00.690 ","End":"07:05.040","Text":"I\u0027ll just get an extra factor of 2 in the denominator."},{"Start":"07:05.040 ","End":"07:10.010","Text":"In other words, I\u0027m going to get the arccosine of 1/2."},{"Start":"07:10.010 ","End":"07:14.865","Text":"The cosine of 1/2 is equal to,"},{"Start":"07:14.865 ","End":"07:16.530","Text":"I know it in degrees,"},{"Start":"07:16.530 ","End":"07:21.059","Text":"it\u0027s 60 degrees, but we\u0027re working in radians,"},{"Start":"07:21.059 ","End":"07:26.030","Text":"so that would make it Pi over 3 because Pi is a 180 degrees."},{"Start":"07:26.030 ","End":"07:30.629","Text":"The integral is from,"},{"Start":"07:31.010 ","End":"07:34.650","Text":"I\u0027ll continue actually down here,"},{"Start":"07:34.650 ","End":"07:38.160","Text":"we get equals meaning,"},{"Start":"07:38.160 ","End":"07:46.970","Text":"for the integral, I\u0027m just going to continue down here rather than moving stuff around."},{"Start":"07:46.970 ","End":"07:53.820","Text":"We get 5 times the integral."},{"Start":"07:53.820 ","End":"07:58.065","Text":"From down here, it\u0027s the 2/5,"},{"Start":"07:58.065 ","End":"07:59.565","Text":"so that\u0027s the 0."},{"Start":"07:59.565 ","End":"08:03.040","Text":"Up here, it\u0027s the Pi over 3."},{"Start":"08:04.220 ","End":"08:11.580","Text":"Some things cancel immediately, 2/5 with 2/5."},{"Start":"08:11.580 ","End":"08:16.650","Text":"1 over cosine t will knock the exponent down by 1,"},{"Start":"08:16.650 ","End":"08:22.285","Text":"so I can say that this will take out this 2."},{"Start":"08:22.285 ","End":"08:27.670","Text":"The 5 with the 5 will go."},{"Start":"08:28.880 ","End":"08:33.010","Text":"Yeah, so I\u0027ll change this into a 2."},{"Start":"08:33.220 ","End":"08:43.530","Text":"Meanwhile, I have absolute value of tangent t times sine"},{"Start":"08:43.530 ","End":"08:53.000","Text":"t over cosine t, dt."},{"Start":"08:53.000 ","End":"08:59.425","Text":"But I\u0027d like to know if this is plus or minus tangent t. I don\u0027t want the absolute value."},{"Start":"08:59.425 ","End":"09:02.480","Text":"The question is; when t,"},{"Start":"09:02.480 ","End":"09:03.950","Text":"t is like an angle,"},{"Start":"09:03.950 ","End":"09:08.179","Text":"my Alpha, goes from 0 to Pi over 3,"},{"Start":"09:08.179 ","End":"09:10.325","Text":"that\u0027s 0 to 60 degrees?"},{"Start":"09:10.325 ","End":"09:12.790","Text":"We\u0027re in the first quadrant."},{"Start":"09:12.790 ","End":"09:15.780","Text":"From 0 to 90, or Pi over 2,"},{"Start":"09:15.780 ","End":"09:16.965","Text":"we\u0027re in the first quadrant."},{"Start":"09:16.965 ","End":"09:19.725","Text":"In the first quadrant the tangent is positive,"},{"Start":"09:19.725 ","End":"09:24.830","Text":"so I can throw out the absolute values."},{"Start":"09:24.830 ","End":"09:27.205","Text":"I can get rid of this and this."},{"Start":"09:27.205 ","End":"09:29.195","Text":"Now I have tangent."},{"Start":"09:29.195 ","End":"09:33.740","Text":"Now tangent is sine over cosine."},{"Start":"09:33.740 ","End":"09:36.230","Text":"What I\u0027m now left with,"},{"Start":"09:36.230 ","End":"09:37.760","Text":"is I throw out the absolute value,"},{"Start":"09:37.760 ","End":"09:41.895","Text":"replace the tangent with sine over cosine,"},{"Start":"09:41.895 ","End":"09:48.800","Text":"and now I have twice the integral from 0 to Pi over 3 of sine"},{"Start":"09:48.800 ","End":"09:58.270","Text":"squared t over cosine squared t dt."},{"Start":"10:00.080 ","End":"10:03.030","Text":"I don\u0027t want to scroll down, I know what I\u0027ll clear,"},{"Start":"10:03.030 ","End":"10:06.860","Text":"the green stuff and I\u0027ll continue up here."},{"Start":"10:06.860 ","End":"10:09.470","Text":"Now we did this one in the previous exercise."},{"Start":"10:09.470 ","End":"10:13.520","Text":"I remember we did the sine squared is 1 minus the cosine squared."},{"Start":"10:13.520 ","End":"10:20.690","Text":"Anyway, we ended up with the tangent t minus t. But in our case,"},{"Start":"10:20.690 ","End":"10:30.410","Text":"there\u0027s also a 2 that we have to carry and the limits are 0 to Pi over 3."},{"Start":"10:30.410 ","End":"10:32.960","Text":"Let\u0027s see what this equals."},{"Start":"10:32.960 ","End":"10:35.750","Text":"If we plug in 0,"},{"Start":"10:35.750 ","End":"10:39.605","Text":"we don\u0027t get anything because tangent of 0 is 0 and this would be 0."},{"Start":"10:39.605 ","End":"10:45.420","Text":"So we just need the Pi over 3 or 60 degrees, you can think of it."},{"Start":"10:45.420 ","End":"10:53.025","Text":"Now the tangent of Pi over 3 is just square root of 3."},{"Start":"10:53.025 ","End":"10:54.930","Text":"You could do it on a calculator,"},{"Start":"10:54.930 ","End":"10:58.890","Text":"but there\u0027s also a table of special angles."},{"Start":"10:59.150 ","End":"11:01.620","Text":"When t is Pi over 3,"},{"Start":"11:01.620 ","End":"11:04.260","Text":"then this is Pi over 3."},{"Start":"11:04.260 ","End":"11:06.380","Text":"Yeah, we could open the brackets,"},{"Start":"11:06.380 ","End":"11:10.160","Text":"but we could just leave the answer like this."},{"Start":"11:10.160 ","End":"11:13.110","Text":"We are done."}],"ID":4409},{"Watched":false,"Name":"Exercise 12","Duration":"4m 48s","ChapterTopicVideoID":6676,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we have to compute the following definite integral."},{"Start":"00:04.110 ","End":"00:09.810","Text":"But this is exactly the same integrand as in the previous exercise."},{"Start":"00:09.810 ","End":"00:12.240","Text":"Only the integration limits have changed,"},{"Start":"00:12.240 ","End":"00:14.160","Text":"so I\u0027m going to use the results from there."},{"Start":"00:14.160 ","End":"00:16.770","Text":"I\u0027d just like to remind you what we did there is we made"},{"Start":"00:16.770 ","End":"00:21.635","Text":"a substitution that x equals 2/5 cosine"},{"Start":"00:21.635 ","End":"00:27.090","Text":"t. The reverse substitution was t equals"},{"Start":"00:27.090 ","End":"00:35.340","Text":"r cosine of 2/5x."},{"Start":"00:35.340 ","End":"00:39.000","Text":"What we got as a result of the integration after"},{"Start":"00:39.000 ","End":"00:45.000","Text":"the substitution was the integral of, let\u0027s see,"},{"Start":"00:45.000 ","End":"00:49.700","Text":"absolute value of tangent of t times"},{"Start":"00:49.700 ","End":"00:55.960","Text":"sine t over cosine t dt."},{"Start":"00:55.960 ","End":"01:00.530","Text":"But the limits will have changed because the limits here have changed."},{"Start":"01:00.530 ","End":"01:04.975","Text":"After we compute the upper and lower limit. Let\u0027s see."},{"Start":"01:04.975 ","End":"01:10.545","Text":"If x is equal to minus 2/5,"},{"Start":"01:10.545 ","End":"01:15.550","Text":"then t equals 2 over this minus 2/5 is just gives minus 2,"},{"Start":"01:15.550 ","End":"01:16.890","Text":"so it\u0027s minus 1."},{"Start":"01:16.890 ","End":"01:22.440","Text":"T is arc cosine of minus 1."},{"Start":"01:22.440 ","End":"01:29.780","Text":"I know this in degrees that it\u0027s 180 degrees is where the cosine is minus 1."},{"Start":"01:29.780 ","End":"01:31.910","Text":"If it\u0027s 180 degrees,"},{"Start":"01:31.910 ","End":"01:35.505","Text":"that makes it equal to Pi."},{"Start":"01:35.505 ","End":"01:38.550","Text":"So the upper limit is Pi."},{"Start":"01:38.550 ","End":"01:42.720","Text":"If x is minus 4/5,"},{"Start":"01:42.720 ","End":"01:47.505","Text":"then we\u0027re going to get an extra factor of 2 in the denominator."},{"Start":"01:47.505 ","End":"01:51.980","Text":"It\u0027s going to be the arc cosine of minus 1/2,"},{"Start":"01:51.980 ","End":"01:54.020","Text":"and for minus 1/2,"},{"Start":"01:54.020 ","End":"01:58.805","Text":"I know it\u0027s a 180 minus 60, 120 degrees,"},{"Start":"01:58.805 ","End":"02:05.025","Text":"that would make it 2/3 of Pi because Pi is a 180."},{"Start":"02:05.025 ","End":"02:10.695","Text":"Now, in this range from 120-180 degrees,"},{"Start":"02:10.695 ","End":"02:15.270","Text":"I know that tangent is negative."},{"Start":"02:15.270 ","End":"02:19.160","Text":"The absolute value of tangent t is going to"},{"Start":"02:19.160 ","End":"02:26.340","Text":"be minus tangent t. I forgot before we had a 2 here."},{"Start":"02:26.340 ","End":"02:30.945","Text":"I can write this as minus 2,"},{"Start":"02:30.945 ","End":"02:34.885","Text":"the same integral, 2Pi/3-Pi,"},{"Start":"02:34.885 ","End":"02:37.535","Text":"but this time without the absolute value."},{"Start":"02:37.535 ","End":"02:40.775","Text":"As before, tangent is sine over cosine,"},{"Start":"02:40.775 ","End":"02:48.480","Text":"so we end up with sine squared t over cosine squared t dt."},{"Start":"02:49.160 ","End":"02:55.915","Text":"Again, as before, we got that this integral was tangent t,"},{"Start":"02:55.915 ","End":"03:06.955","Text":"the indefinite integral minus t. But now I need to take it between 2Pi over 3 and Pi,"},{"Start":"03:06.955 ","End":"03:10.165","Text":"and that will make it minus 2."},{"Start":"03:10.165 ","End":"03:12.610","Text":"Now let\u0027s see if I put Pi,"},{"Start":"03:12.610 ","End":"03:17.450","Text":"tangent of Pi is equal to 0,"},{"Start":"03:17.450 ","End":"03:27.315","Text":"so that would make it minus 2 times 0 minus Pi for the Pi,"},{"Start":"03:27.315 ","End":"03:31.680","Text":"and then less the same thing with 2Pi/3."},{"Start":"03:31.680 ","End":"03:41.540","Text":"The tangent of 2Pi/3 is minus the square root of 3 because it\u0027s a 120 degrees,"},{"Start":"03:41.540 ","End":"03:44.045","Text":"so it\u0027s minus the tangent of 60 degrees,"},{"Start":"03:44.045 ","End":"03:45.875","Text":"which is square root of 3."},{"Start":"03:45.875 ","End":"03:52.040","Text":"This would be minus the square root of 3 minus 2Pi/3."},{"Start":"03:52.040 ","End":"03:55.400","Text":"Hope I\u0027m not making a mistake here."},{"Start":"03:55.400 ","End":"03:57.200","Text":"Let\u0027s see what this becomes."},{"Start":"03:57.200 ","End":"03:58.550","Text":"It\u0027s minus 2."},{"Start":"03:58.550 ","End":"04:03.480","Text":"Now here we have minus Pi and here everything becomes plus,"},{"Start":"04:03.480 ","End":"04:10.430","Text":"plus square root of 3 plus 2Pi/3."},{"Start":"04:10.430 ","End":"04:14.705","Text":"This equals now minus 2 times minus Pi will be"},{"Start":"04:14.705 ","End":"04:19.650","Text":"2Pi minus twice square root"},{"Start":"04:19.650 ","End":"04:26.610","Text":"of 3 minus 4Pi/3."},{"Start":"04:26.610 ","End":"04:30.945","Text":"Now, 2 minus 4/3 is 2/3."},{"Start":"04:30.945 ","End":"04:34.170","Text":"It\u0027s 2 minus 1/3, which is 2/3."},{"Start":"04:34.170 ","End":"04:44.295","Text":"Ultimately I get 2/3Pi minus twice the square root of 3."},{"Start":"04:44.295 ","End":"04:49.600","Text":"That\u0027s the answer I get anyway, and we\u0027re done."}],"ID":6737},{"Watched":false,"Name":"Exercise 13","Duration":"5m 37s","ChapterTopicVideoID":6677,"CourseChapterTopicPlaylistID":3683,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"Here we have to compute the following integral,"},{"Start":"00:02.610 ","End":"00:05.025","Text":"which I\u0027ve copied over here."},{"Start":"00:05.025 ","End":"00:08.130","Text":"The usual thing is to try a substitution t equals"},{"Start":"00:08.130 ","End":"00:11.265","Text":"square root of 9 minus x squared, but it doesn\u0027t work."},{"Start":"00:11.265 ","End":"00:13.860","Text":"At this point, you\u0027re sophisticated enough to know"},{"Start":"00:13.860 ","End":"00:16.725","Text":"that it\u0027s 1 of these trigonometric substitutions,"},{"Start":"00:16.725 ","End":"00:22.440","Text":"and particularly of the form square root of a squared minus x squared,"},{"Start":"00:22.440 ","End":"00:24.480","Text":"where in this case 9 is 3 squared,"},{"Start":"00:24.480 ","End":"00:26.340","Text":"so a is 3."},{"Start":"00:26.340 ","End":"00:29.475","Text":"These are the equations of this substitution."},{"Start":"00:29.475 ","End":"00:30.900","Text":"Let\u0027s just write what they mean."},{"Start":"00:30.900 ","End":"00:32.460","Text":"In our particular case,"},{"Start":"00:32.460 ","End":"00:33.960","Text":"we have 1, 2, and 3."},{"Start":"00:33.960 ","End":"00:37.130","Text":"It just mean that we replace a with 3."},{"Start":"00:37.130 ","End":"00:42.595","Text":"We are substituting x equals 3 sine t and"},{"Start":"00:42.595 ","End":"00:49.765","Text":"the reverse substitution is that t equals arcsine of x over 3."},{"Start":"00:49.765 ","End":"00:56.060","Text":"The second equation tells us that the square root of 9 minus x"},{"Start":"00:56.060 ","End":"01:03.090","Text":"squared is equal to 3 cosine t. The last 1,"},{"Start":"01:03.090 ","End":"01:09.690","Text":"that dx is equal to 3 cosine t dt."},{"Start":"01:09.690 ","End":"01:12.110","Text":"Armed with these 3,"},{"Start":"01:12.110 ","End":"01:19.275","Text":"we can now tackle this 1 and say that this is equal to the integral of,"},{"Start":"01:19.275 ","End":"01:22.920","Text":"let us see now, I\u0027ll start with dx."},{"Start":"01:22.920 ","End":"01:25.435","Text":"Dx is 3 cosine t dt,"},{"Start":"01:25.435 ","End":"01:28.700","Text":"so it\u0027s 3 cosine t. There\u0027s going to be"},{"Start":"01:28.700 ","End":"01:32.495","Text":"a dividing line and let me put the dt at the side."},{"Start":"01:32.495 ","End":"01:35.780","Text":"That takes care of dx."},{"Start":"01:35.780 ","End":"01:38.270","Text":"Now let\u0027s do the next easiest one."},{"Start":"01:38.270 ","End":"01:44.120","Text":"The square root of 9 minus x squared is 3 cosine t. Here we have a"},{"Start":"01:44.120 ","End":"01:51.595","Text":"3 cosine t. Lastly, we have x^4."},{"Start":"01:51.595 ","End":"01:53.310","Text":"If x is this,"},{"Start":"01:53.310 ","End":"01:56.590","Text":"then we just have here,"},{"Start":"01:57.080 ","End":"02:06.530","Text":"3^4 sine^4 of t. Let\u0027s simplify."},{"Start":"02:06.530 ","End":"02:10.580","Text":"This is equal to the integral."},{"Start":"02:10.580 ","End":"02:14.075","Text":"Let\u0027s see. Some things may cancel."},{"Start":"02:14.075 ","End":"02:17.349","Text":"This 3 certainly goes with this 3,"},{"Start":"02:17.349 ","End":"02:20.265","Text":"the whole 3 cosine t cancels."},{"Start":"02:20.265 ","End":"02:23.850","Text":"What I\u0027m left with is 1 over 3^4,"},{"Start":"02:23.850 ","End":"02:26.925","Text":"which is 1 over 81,"},{"Start":"02:26.925 ","End":"02:34.475","Text":"and I\u0027m left with 1 over sine^4 t dt,"},{"Start":"02:34.475 ","End":"02:36.500","Text":"which is a trigonometric integral."},{"Start":"02:36.500 ","End":"02:41.030","Text":"I\u0027m going to have to use a bag of tricks and trigonometrical identities on this."},{"Start":"02:41.030 ","End":"02:46.955","Text":"What I will do is write this integral as 1 over"},{"Start":"02:46.955 ","End":"02:56.980","Text":"sine squared t times sine squared t. I\u0027ll split the power of 4 into 2 and 2."},{"Start":"02:56.980 ","End":"03:07.590","Text":"Now, I\u0027m going to use the identity that 1 over sine squared is cotangent squared."},{"Start":"03:08.150 ","End":"03:13.005","Text":"Here I have over sine squared,"},{"Start":"03:13.005 ","End":"03:20.075","Text":"and now I\u0027m going to make another substitution and let u equals cotangent squared of"},{"Start":"03:20.075 ","End":"03:27.960","Text":"t. I also forgot to write the plus 1 here."},{"Start":"03:27.960 ","End":"03:37.235","Text":"Forgive. U equals cotangent t and du is the derivative of cotangent,"},{"Start":"03:37.235 ","End":"03:42.535","Text":"which is minus 1 over sine squared t dt,"},{"Start":"03:42.535 ","End":"03:44.940","Text":"which is almost what we have."},{"Start":"03:44.940 ","End":"03:47.045","Text":"If we continue here,"},{"Start":"03:47.045 ","End":"03:52.810","Text":"we will get 1 over 81 times the integral."},{"Start":"03:52.810 ","End":"03:56.570","Text":"I\u0027ll make this a minus or put the minus all the way here and"},{"Start":"03:56.570 ","End":"04:01.610","Text":"then the dt over minus sine squared t will be du."},{"Start":"04:01.610 ","End":"04:03.350","Text":"What we have here, basically,"},{"Start":"04:03.350 ","End":"04:08.610","Text":"is just u squared plus 1 du."},{"Start":"04:08.610 ","End":"04:11.120","Text":"That\u0027s easy enough to do."},{"Start":"04:11.120 ","End":"04:14.689","Text":"That is going to equal minus"},{"Start":"04:14.689 ","End":"04:22.680","Text":"1 over 81 times u cubed over 3 plus u,"},{"Start":"04:22.680 ","End":"04:27.980","Text":"so it\u0027s just plus C. But now I have to start substituting"},{"Start":"04:27.980 ","End":"04:34.885","Text":"back because I have to get from u back to t and from t back to x. U is cotangent t,"},{"Start":"04:34.885 ","End":"04:40.670","Text":"so this is equal to minus 1 over 81 of"},{"Start":"04:40.670 ","End":"04:50.610","Text":"cotangent cubed of t over 3 plus cotangent t,"},{"Start":"04:50.610 ","End":"04:52.935","Text":"all this plus a constant."},{"Start":"04:52.935 ","End":"04:56.460","Text":"Now I have to substitute back from here,"},{"Start":"04:56.460 ","End":"05:00.280","Text":"that t is arcsine of x over 3."},{"Start":"05:00.680 ","End":"05:05.040","Text":"This is equal to minus 1 over"},{"Start":"05:05.040 ","End":"05:12.420","Text":"81 times 1/3 of cotangent t cubed,"},{"Start":"05:12.420 ","End":"05:16.950","Text":"which is cotangent cubed of t,"},{"Start":"05:16.950 ","End":"05:20.894","Text":"which is arcsine of x over 3,"},{"Start":"05:20.894 ","End":"05:29.015","Text":"plus cotangent of arcsine of x over 3 because that\u0027s this t,"},{"Start":"05:29.015 ","End":"05:32.840","Text":"arcsine of x over 3,"},{"Start":"05:32.840 ","End":"05:38.130","Text":"and at the end, plus C. That\u0027s it."}],"ID":6738}],"Thumbnail":null,"ID":3683}]

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