WEBVTT 00:00:01.340 --> 00:00:04.616 Sometimes integrals involving trigonometric functions can be 00:00:04.616 --> 00:00:08.828 evaluated by first of all using trigonometric identities to 00:00:08.828 --> 00:00:13.040 rewrite the integrand. That's the quantity we're trying to 00:00:13.040 --> 00:00:18.188 integrate an alternative form, which is a bit more amenable to 00:00:18.188 --> 00:00:21.645 integration. Sometimes a trigonometric substitution is 00:00:21.645 --> 00:00:26.175 appropriate. Both of these techniques we look at in this 00:00:26.175 --> 00:00:31.034 unit. Before we start I want to give you a couple of preliminary 00:00:31.034 --> 00:00:35.025 results which will be using over and over again and which will be 00:00:35.025 --> 00:00:39.323 very important and the first one is I want you to make sure that 00:00:39.323 --> 00:00:43.007 you know that the integral of the cosine of a constant times 00:00:43.007 --> 00:00:45.768 X. With respect to X. 00:00:46.290 --> 00:00:49.699 Is equal to one over that constant. 00:00:50.850 --> 00:00:55.228 Multiplied by the sign of KX plus a constant of integration 00:00:55.228 --> 00:00:59.606 as a very important result. If you integrate the cosine, you 00:00:59.606 --> 00:01:00.800 get a sign. 00:01:01.970 --> 00:01:05.415 And if there's a constant in front of the X that appears down 00:01:05.415 --> 00:01:08.595 here will take that as read in all the examples which follow. 00:01:09.360 --> 00:01:15.245 Another important results is the integral of a sign. The integral 00:01:15.245 --> 00:01:22.735 of sine KX with respect to X is minus one over K cosine KX. 00:01:23.390 --> 00:01:26.950 Plus a constant we're integrating assign. The result 00:01:26.950 --> 00:01:31.400 is minus the cosine and the constant factor. There appears 00:01:31.400 --> 00:01:34.515 out down here as well, so those 00:01:34.515 --> 00:01:37.300 two results. Very important. 00:01:37.900 --> 00:01:40.720 You should have them at your fingertips and we can call upon 00:01:40.720 --> 00:01:43.305 them whenever we want them in the rest of the video. 00:01:44.090 --> 00:01:47.897 We also want to call appan trigonometric identity's. I'm 00:01:47.897 --> 00:01:52.127 going to assume that you've seen a lot of trigonometric 00:01:52.127 --> 00:01:55.382 identities before. We have a table of trigonometric 00:01:55.382 --> 00:01:58.649 identities here, such as the table that you might have seen 00:01:58.649 --> 00:02:01.916 many times before. If you want this specific table, you'll find 00:02:01.916 --> 00:02:03.401 it in the printed notes 00:02:03.401 --> 00:02:07.022 accompanying the video. Why might we want to use 00:02:07.022 --> 00:02:09.980 trigonometric identities? Well, for example, we've just 00:02:09.980 --> 00:02:14.108 seen that we already know how to integrate the sign of a 00:02:14.108 --> 00:02:17.892 quantity and the cosine of the quantity. But suppose we want 00:02:17.892 --> 00:02:21.332 to integrate assign multiplied by a cosine or cosine times 00:02:21.332 --> 00:02:24.428 cosine or assigned times assign. We don't actually know 00:02:24.428 --> 00:02:27.868 how to do those integrals. Integrals at the moment, but 00:02:27.868 --> 00:02:30.620 if we use trigonometric identities, we can rewrite 00:02:30.620 --> 00:02:34.404 these in terms of just single sine and cosine terms, which 00:02:34.404 --> 00:02:35.780 we can then integrate. 00:02:36.900 --> 00:02:39.665 Also, the trigonometric identities identities allow us 00:02:39.665 --> 00:02:44.010 to integrate powers of sines and cosines. You'll see that using 00:02:44.010 --> 00:02:47.960 these identity's? We've got powers of cosine powers of sign 00:02:47.960 --> 00:02:53.095 and the identity is allow us to write into grams in terms of 00:02:53.095 --> 00:02:55.070 cosines and sines of double 00:02:55.070 --> 00:02:59.070 angles. We know how to integrate these already using the results. 00:02:59.070 --> 00:03:02.646 I've just reminded you of, so I'm going to assume that you've 00:03:02.646 --> 00:03:06.222 got a table like this at your fingertips, and we can call 00:03:06.222 --> 00:03:08.010 appan it whenever we need to. 00:03:08.500 --> 00:03:12.472 OK, let's have a look at the first example and the example 00:03:12.472 --> 00:03:16.444 that I'm going to look at is a definite integral. The integral 00:03:16.444 --> 00:03:22.402 from X is not to X is π of the sine squared of X DX. So note in 00:03:22.402 --> 00:03:25.712 particular, we've gotta power here. We're looking at the sign 00:03:25.712 --> 00:03:30.346 squared of X. What I'm going to do is go back to the table. 00:03:31.070 --> 00:03:34.931 And look for an identity that will allow us to change the sign 00:03:34.931 --> 00:03:38.495 squared X into something else. Let me just flip back to the 00:03:38.495 --> 00:03:39.386 table of trigonometric 00:03:39.386 --> 00:03:43.960 identities. The identity that I'm going to use this one, the 00:03:43.960 --> 00:03:45.079 cosine of 2A. 00:03:45.650 --> 00:03:48.618 Is 1 minus twice sign square day? 00:03:49.650 --> 00:03:52.884 If you inspect this carefully, you'll see that this will enable 00:03:52.884 --> 00:03:56.706 us to change a sine squared into the cosine of a double angle. 00:03:57.690 --> 00:04:00.576 Let me write that down again. 00:04:00.580 --> 00:04:07.396 Cosine of 2 A is equal to 1 00:04:07.396 --> 00:04:10.804 minus twice sign squared 00:04:10.804 --> 00:04:13.990 A. First of all, I'm going to rearrange this to get 00:04:13.990 --> 00:04:15.155 sine squared on its own. 00:04:16.260 --> 00:04:20.730 If we add two sine squared data both sides, then I can get it on 00:04:20.730 --> 00:04:27.060 this side. And if I subtract cosine 2A from both sides, are 00:04:27.060 --> 00:04:29.665 remove it from the left. 00:04:29.790 --> 00:04:37.190 Finally, if I divide both sides by two, I'll be 00:04:37.190 --> 00:04:40.890 left with sine squared A. 00:04:40.900 --> 00:04:46.640 And this is the result that I want to use to help me to 00:04:46.640 --> 00:04:51.560 evaluate this integral because of what it will allow me to do. 00:04:51.560 --> 00:04:56.890 Is it will allow me to change a quantity involving the square of 00:04:56.890 --> 00:05:00.990 a trig function into a quantity involving double angles. So 00:05:00.990 --> 00:05:03.450 let's use it in this case. 00:05:04.410 --> 00:05:10.470 The integral will become the integral from note to pie. 00:05:11.340 --> 00:05:18.446 Sine squared X using this formula will be 1 minus cosine 00:05:18.446 --> 00:05:22.266 twice X. All divided by 00:05:22.266 --> 00:05:24.710 two. Integrated with respect to X. 00:05:26.120 --> 00:05:31.632 I've taken out the fact that 1/2 here and I'm left with the 00:05:31.632 --> 00:05:36.720 numerator 1 minus cosine 2X to be integrated with respect to X. 00:05:37.620 --> 00:05:39.120 This is straightforward to 00:05:39.120 --> 00:05:43.884 finish off. So definite integral. So I have square 00:05:43.884 --> 00:05:49.452 brackets. The integral of one with respect to X is simply X. 00:05:50.470 --> 00:05:55.030 And the integral of cosine 2 X we know from our preliminary 00:05:55.030 --> 00:06:00.350 work is just going to be sine 2X divided by two with a minus 00:06:00.350 --> 00:06:03.770 sign there and the limits are not and pie. 00:06:05.770 --> 00:06:10.424 We finish this off by first of all, putting the upper limit in, 00:06:10.424 --> 00:06:14.362 so we want X replaced by pie here and pie here. 00:06:15.070 --> 00:06:17.620 The sign of 2π is 0. 00:06:18.520 --> 00:06:22.645 So when we put the upper limit in will just get. 00:06:22.930 --> 00:06:26.320 Pie by substituting for X here. 00:06:27.030 --> 00:06:30.880 Let me put the lower limit in. 00:06:30.880 --> 00:06:33.547 X being not will be 0 here. 00:06:34.210 --> 00:06:38.830 And sign of note here, which is not so both of those terms will 00:06:38.830 --> 00:06:43.120 become zero when we put the lower limit in and so we're just 00:06:43.120 --> 00:06:45.760 left with simply 1/2 of Π or π 00:06:45.760 --> 00:06:51.040 by 2. And that's our first example of how we've used a 00:06:51.040 --> 00:06:53.686 trigonometric identity to rewrite an integrand involving 00:06:53.686 --> 00:06:58.222 powers of a trig function in terms of double angles, which we 00:06:58.222 --> 00:07:00.112 already know how to integrate. 00:07:01.360 --> 00:07:09.190 Let's have a look at another example. Suppose we want 00:07:09.190 --> 00:07:17.020 to integrate the sign of three X multiplied by the 00:07:17.020 --> 00:07:20.152 cosine of 2 X. 00:07:20.160 --> 00:07:21.248 With respect to X. 00:07:21.770 --> 00:07:26.306 Now we already know how to integrate signs. We know how to 00:07:26.306 --> 00:07:30.464 integrate cosines, but we have a problem here because there's a 00:07:30.464 --> 00:07:34.244 product. These two terms are multiplied together and we don't 00:07:34.244 --> 00:07:35.756 know how to proceed. 00:07:36.590 --> 00:07:41.282 What we do is look in our table of trigonometric identities for 00:07:41.282 --> 00:07:45.192 an example where we've gotta sign multiplied by a cosine. 00:07:45.192 --> 00:07:47.538 Let's go back to the table. 00:07:47.560 --> 00:07:53.743 The first entry in our table involves assign multiplied 00:07:53.743 --> 00:07:55.804 by a cosine. 00:07:56.660 --> 00:08:02.048 Let me write this formula down again. 2 sign a cosine be. 00:08:02.050 --> 00:08:07.690 Is equal 00:08:07.690 --> 00:08:14.302 to. The sign of the sum of A and be 00:08:14.302 --> 00:08:16.450 added to the sign of the 00:08:16.450 --> 00:08:23.935 difference A-B. And this is the identity that I 00:08:23.935 --> 00:08:31.655 will use in order to rewrite this integrand 00:08:31.655 --> 00:08:39.375 as two separate integrals. We identify the A's 00:08:39.375 --> 00:08:43.280 3X. The B is 2 X. 00:08:44.050 --> 00:08:48.714 The factor of 2 here isn't a problem. We can divide 00:08:48.714 --> 00:08:50.410 everything through by two. 00:08:50.420 --> 00:08:51.785 So we lose it from this side. 00:08:52.770 --> 00:08:57.973 So our integral? What will it become? Well, the integral of 00:08:57.973 --> 00:09:03.649 sign 3X cosine 2X DX will become. We want the integral of 00:09:03.649 --> 00:09:06.960 the sign of the sum of A&B. 00:09:07.540 --> 00:09:12.244 Well, there's some of A&B will be 3X plus 2X, which is 5X. So 00:09:12.244 --> 00:09:14.260 we want the sign of 5X. 00:09:15.520 --> 00:09:19.502 Added to the sign of the difference of amb. Well a 00:09:19.502 --> 00:09:23.846 being 3X B being 2X A-B will be 3X subtract 2 X 00:09:23.846 --> 00:09:28.552 which is just One X. So we want the sign of X all 00:09:28.552 --> 00:09:32.172 divided by two and we want to integrate that with 00:09:32.172 --> 00:09:33.258 respect to X. 00:09:34.510 --> 00:09:38.995 So what have we done? We've used the trig identity to change the 00:09:38.995 --> 00:09:43.135 product of a signing cosine into the sum of two separate sign 00:09:43.135 --> 00:09:46.585 terms, which we can integrate straight away. We can integrate 00:09:46.585 --> 00:09:48.655 that taking the factor of 1/2 00:09:48.655 --> 00:09:56.314 out. The integral of sign 5X will be minus the cosine of 5X 00:09:56.314 --> 00:09:58.000 divided by 5. 00:09:58.740 --> 00:10:03.628 And the integral of sine X will be just minus cosine X, and 00:10:03.628 --> 00:10:05.508 they'll be a constant of 00:10:05.508 --> 00:10:10.924 integration. And just to tidy it up, at the end we're going to 00:10:10.924 --> 00:10:14.968 have minus the half with the five at the bottom. There will 00:10:14.968 --> 00:10:18.001 give you minus cosine 5X all divided by 10. 00:10:19.060 --> 00:10:23.092 And there's a half with this term here, so it's minus cosine 00:10:23.092 --> 00:10:24.436 X divided by two. 00:10:25.220 --> 00:10:27.950 Plus a constant of integration. 00:10:28.500 --> 00:10:30.288 And that's the solution of this 00:10:30.288 --> 00:10:35.420 problem. Let's explore the integral of products of sines 00:10:35.420 --> 00:10:41.900 and cosines a little bit further, and what I want to look 00:10:41.900 --> 00:10:48.920 at now is integrals of the form the integral of sign to the 00:10:48.920 --> 00:10:54.320 power MX multiplied by cosine to the power NX DX. 00:10:54.930 --> 00:10:58.230 Well, look at a whole family of integrals like this, but in 00:10:58.230 --> 00:11:01.805 particular for the first example I'm going to look at the case of 00:11:01.805 --> 00:11:03.730 what happens when M is an odd 00:11:03.730 --> 00:11:09.300 number. Whenever you have an integral like this, when M is 00:11:09.300 --> 00:11:14.239 odd, the following process will work. Let's look at a specific 00:11:14.239 --> 00:11:18.280 case, supposing I want to integrate sine cubed X. 00:11:18.820 --> 00:11:23.500 Multiplied by cosine squared XDX. 00:11:24.590 --> 00:11:27.530 Notice that M. 00:11:28.220 --> 00:11:30.670 Is an odd number and is 3. 00:11:31.560 --> 00:11:35.356 There's a little trick here that we're going to do now, and it's 00:11:35.356 --> 00:11:38.860 the sort of trick that comes with practice and seeing lots of 00:11:38.860 --> 00:11:42.364 examples. What we're going to do is we're going to rewrite the 00:11:42.364 --> 00:11:44.116 sign cubed X in a slightly 00:11:44.116 --> 00:11:49.285 different form. We're going to recognize that sign cubed can be 00:11:49.285 --> 00:11:53.290 written as sine squared X multiplied by Sign X. 00:11:53.800 --> 00:11:57.568 That's a little trick. The sign cubed can be written as sine 00:11:57.568 --> 00:12:01.410 squared times sign. So our integral can be 00:12:01.410 --> 00:12:05.202 written as sine squared X times sign X 00:12:05.202 --> 00:12:08.046 multiplied by cosine squared X DX. 00:12:09.240 --> 00:12:12.670 And then I'm going to pick a trigonometric identity involving 00:12:12.670 --> 00:12:16.443 sine squared to write it in terms of cosine squared. Let's 00:12:16.443 --> 00:12:17.472 find that identity. 00:12:18.090 --> 00:12:21.115 With an identity here, which says that sine squared of an 00:12:21.115 --> 00:12:22.765 angle plus cost squared of an 00:12:22.765 --> 00:12:27.834 angle is one. If we rearrange this, we can write that sine 00:12:27.834 --> 00:12:32.722 squared of an angle is 1 minus the cosine squared of an angle 00:12:32.722 --> 00:12:33.850 will use that. 00:12:34.700 --> 00:12:38.596 Sine squared of any 00:12:38.596 --> 00:12:44.860 angle. Is equal to 1 minus the cosine squared over any 00:12:44.860 --> 00:12:51.290 angle. Will use that in here to change the sign squared X into 00:12:51.290 --> 00:12:55.740 terms involving cosine squared X. Let's see what happens. This 00:12:55.740 --> 00:13:00.190 integral will become the integral of or sign squared X. 00:13:00.790 --> 00:13:03.500 Will become one minus cosine 00:13:03.500 --> 00:13:09.320 squared X. There's still the terms cynex. 00:13:11.620 --> 00:13:13.657 And at the end we still got 00:13:13.657 --> 00:13:17.378 cosine squared X. Now this is looking a bit complicated, but 00:13:17.378 --> 00:13:20.641 as we'll see it's all going to come out in the Wash. Let's 00:13:20.641 --> 00:13:22.147 remove the brackets here and see 00:13:22.147 --> 00:13:27.251 what we've got. There's a one multiplied by all this sign X 00:13:27.251 --> 00:13:28.767 times cosine squared X. 00:13:29.440 --> 00:13:33.265 So that's just sign X times cosine squared X 00:13:33.265 --> 00:13:37.090 will want to integrate that with respect to X. 00:13:38.510 --> 00:13:42.443 There's also cosine squared X multiplied by all this. 00:13:42.980 --> 00:13:47.468 Now the cosine squared X with this cosine squared X will give 00:13:47.468 --> 00:13:50.086 us a cosine, so the power 4X. 00:13:51.840 --> 00:13:53.920 There's also the sign X. 00:13:54.850 --> 00:13:56.800 And we want to integrate that. 00:13:57.380 --> 00:14:00.851 Also, with respect to X and there was a minus sign in front, 00:14:00.851 --> 00:14:02.720 so that's going to go in there. 00:14:03.350 --> 00:14:05.702 So we've expanded the brackets here and written. 00:14:05.702 --> 00:14:07.172 This is 2 separate integrals. 00:14:08.410 --> 00:14:13.591 Now, each of these integrals can be evaluated by making a 00:14:13.591 --> 00:14:18.301 substitution. If we make a substitution and let you equals 00:14:18.301 --> 00:14:24.168 cosine X. The differential du is du DX. 00:14:24.970 --> 00:14:30.880 DX Do you DX if we differentiate cosine, X will get 00:14:30.880 --> 00:14:32.564 minus the sign X. 00:14:33.110 --> 00:14:36.582 So we've got du is minus sign X 00:14:36.582 --> 00:14:42.560 DX. Now look at what we've got when we make this substitution. 00:14:42.560 --> 00:14:48.020 The cosine squared X will become simply you squared and sign X DX 00:14:48.020 --> 00:14:53.060 altogether can be written as a minus du, so this will become. 00:14:53.830 --> 00:14:55.318 Minus the integral. 00:14:56.010 --> 00:14:57.420 Of you squared. 00:14:58.010 --> 00:14:58.940 Do you? 00:15:01.250 --> 00:15:06.398 What about this term? We've got cosine to the power four cosine 00:15:06.398 --> 00:15:09.830 to the power 4X will be you to 00:15:09.830 --> 00:15:15.240 the powerful. And sign X DX sign X DX is minus DU. 00:15:15.240 --> 00:15:18.425 There's another minus sign here, so overall 00:15:18.425 --> 00:15:22.520 will have plus the integral of you to the 00:15:22.520 --> 00:15:23.885 four, do you? 00:15:25.450 --> 00:15:29.641 Now these are very very simple integrals to finish the integral 00:15:29.641 --> 00:15:32.308 of you squared is you cubed over 00:15:32.308 --> 00:15:38.630 3? The integral of you to the four is due to the five over 5 00:15:38.630 --> 00:15:40.455 plus a constant of integration. 00:15:42.470 --> 00:15:48.008 All we need to do to finish off is return to our original 00:15:48.008 --> 00:15:53.120 variables. Remember, you was cosine of X, so we finish off by 00:15:53.120 --> 00:15:54.398 writing minus 1/3. 00:15:54.970 --> 00:15:59.304 You being cosine X means that we've got cosine cubed X. 00:16:00.670 --> 00:16:07.180 Plus 1/5. You to the five will be Co sign 00:16:07.180 --> 00:16:09.100 to the power 5X. 00:16:10.260 --> 00:16:11.760 Plus a constant of integration. 00:16:12.570 --> 00:16:17.190 And that's the solution to the problem that we started with. 00:16:18.220 --> 00:16:24.184 Let's stick with the same sort of family of integrals, so we're 00:16:24.184 --> 00:16:30.148 still sticking with the integral of sign to the power MX cosine 00:16:30.148 --> 00:16:32.633 to the power NX DX. 00:16:33.210 --> 00:16:37.650 And now I'm going to have a look at what happens in the case when 00:16:37.650 --> 00:16:39.130 M is an even number. 00:16:39.650 --> 00:16:42.560 And N is an odd number. 00:16:44.480 --> 00:16:47.252 This method will always work when M is even. An is odd. 00:16:47.790 --> 00:16:52.266 Let's look at a specific case. Suppose we want to integrate the 00:16:52.266 --> 00:16:54.131 sign to the power 4X. 00:16:55.190 --> 00:16:57.749 Cosine cubed X. 00:16:58.350 --> 00:16:59.290 DX 00:17:01.840 --> 00:17:07.118 Notice that M the power of sign is now even em is full. 00:17:08.430 --> 00:17:12.690 And N which is the power of cosine, is odd an IS3. 00:17:13.430 --> 00:17:17.343 What I'm going to do is I'm going to use the identity that 00:17:17.343 --> 00:17:21.256 cosine squared of an angle is 1 minus sign squared of an angle 00:17:21.256 --> 00:17:25.169 and you'll be able to lift that directly from the table we had 00:17:25.169 --> 00:17:28.179 at the beginning, which stated the very important and well 00:17:28.179 --> 00:17:31.189 known results that cosine squared of an angle plus the 00:17:31.189 --> 00:17:34.199 sine squared of an angle is always equal to 1. 00:17:34.890 --> 00:17:40.112 What I'm going to do is I'm going to use this to rewrite the 00:17:40.112 --> 00:17:42.850 cosine term. In here, in terms 00:17:42.850 --> 00:17:47.290 of signs. First of all, I'm going to apply the little trick 00:17:47.290 --> 00:17:53.284 we had before. And split the cosine turn up like this cosine 00:17:53.284 --> 00:17:56.310 cubed. I'm going to write this 00:17:56.310 --> 00:17:59.280 cosine squared. Multiplied by 00:17:59.280 --> 00:18:05.400 cosine. So I've changed the cosine cubed to these two terms 00:18:05.400 --> 00:18:12.220 here. Now I can use the identity to change cosine 00:18:12.220 --> 00:18:15.370 squared X into terms involving 00:18:15.370 --> 00:18:20.558 sine squared. So the integral will become the integral of 00:18:20.558 --> 00:18:23.248 sign. To the power 4X. 00:18:24.060 --> 00:18:30.000 Cosine squared X. We can write as one minus sign, squared X. 00:18:31.460 --> 00:18:35.940 And there's still this term cosine X here as well. 00:18:37.730 --> 00:18:41.305 And all that has to be integrated with respect to X. 00:18:44.240 --> 00:18:48.981 Let me remove the brackets here. When we remove the brackets, 00:18:48.981 --> 00:18:54.584 there will be signed to the 4th X Times one all multiplied by 00:18:54.584 --> 00:19:01.685 cosine X. That'll be signed to the 4th X 00:19:01.685 --> 00:19:05.185 multiplied by sign squared 00:19:05.185 --> 00:19:10.676 X. Which is signed to the 6X or multiplied by cosine X. 00:19:12.130 --> 00:19:19.020 And there's a minus sign in the middle, and we want to integrate 00:19:19.020 --> 00:19:23.150 all that. With respect to X. 00:19:25.340 --> 00:19:29.100 Again, a simple substitution will allow us to finish this 00:19:29.100 --> 00:19:30.980 off. If we let you. 00:19:31.510 --> 00:19:33.250 Be sign X. 00:19:34.350 --> 00:19:35.970 So do you. 00:19:36.480 --> 00:19:39.140 Is cosine X DX. 00:19:39.790 --> 00:19:43.563 This will become immediately the integral of well signed to the 00:19:43.563 --> 00:19:48.022 4th X sign to the 4th X will be you to the four. 00:19:48.780 --> 00:19:54.126 The cosine X times the DX cosine X DX becomes du. 00:19:55.720 --> 00:20:02.520 Subtract. Sign to the six, X will become you to 00:20:02.520 --> 00:20:09.386 the six. And the cosine X DX is du. 00:20:09.470 --> 00:20:13.199 So what we've achieved are two very simple integrals that we 00:20:13.199 --> 00:20:14.894 can complete to finish the 00:20:14.894 --> 00:20:20.754 problem. The integral of you to the four is due to the five over 00:20:20.754 --> 00:20:25.596 5. The integral of you to the six is due to the 7 over 7. 00:20:26.460 --> 00:20:27.549 Plus a constant. 00:20:29.010 --> 00:20:33.690 And then just to finish off, we return to the original variables 00:20:33.690 --> 00:20:37.980 and replace EU with sign X, which will give us 1/5. 00:20:38.510 --> 00:20:42.360 Sign next to the five or sign to the power 5X. 00:20:44.160 --> 00:20:45.090 Minus. 00:20:46.110 --> 00:20:52.440 One 7th. You to the Seven will be signed to the 7X. 00:20:53.050 --> 00:20:56.270 Plus a constant of integration. 00:20:58.300 --> 00:21:01.708 So that's how we deal with integrals of this family. In the 00:21:01.708 --> 00:21:05.968 case when M is an even number and when N is an odd number. Now 00:21:05.968 --> 00:21:09.660 in the case when both M&N are even, you should try using the 00:21:09.660 --> 00:21:13.068 double angle formulas, and I'm not going to do an example of 00:21:13.068 --> 00:21:16.760 that because there isn't time in this video to do that. But there 00:21:16.760 --> 00:21:19.600 are examples in the exercises accompanying the video and you 00:21:19.600 --> 00:21:21.020 should try those for yourself. 00:21:21.730 --> 00:21:28.610 I'm not going to look at some integrals for which 00:21:28.610 --> 00:21:31.362 a trigonometric substitution is 00:21:31.362 --> 00:21:36.787 appropriate. Suppose we want to evaluate this integral. 00:21:36.790 --> 00:21:43.102 The integral of 1 / 1 00:21:43.102 --> 00:21:46.258 plus X squared. 00:21:47.030 --> 00:21:48.178 With respect to X. 00:21:49.710 --> 00:21:53.103 Now the trigonometric substitution that I want to use 00:21:53.103 --> 00:21:59.135 is this one. I want to let X be the tangent of a new variable, X 00:21:59.135 --> 00:22:00.266 equals 10 theater. 00:22:00.920 --> 00:22:04.115 While I picked this particular substitution well, all will 00:22:04.115 --> 00:22:09.085 become clear in time, but I want to just look ahead a little bit 00:22:09.085 --> 00:22:11.215 by letting X equal 10 theater. 00:22:11.750 --> 00:22:14.837 What will have at the denominator down here is 00:22:14.837 --> 00:22:16.552 1 + 10 squared theater. 00:22:17.570 --> 00:22:22.946 One plus X squared will become 1 + 10 squared and we have an 00:22:22.946 --> 00:22:27.170 identity already which says that 1 + 10 squared of an 00:22:27.170 --> 00:22:31.394 angle is equal to the sequence squared of the angle. That's 00:22:31.394 --> 00:22:36.002 an identity that we had on the table right at the beginning, 00:22:36.002 --> 00:22:40.610 so the idea is that by making this substitution, 1 + 10 00:22:40.610 --> 00:22:44.450 squared can be replaced by a single term sequence squared, 00:22:44.450 --> 00:22:47.522 as we'll see, so let's progress with that 00:22:47.522 --> 00:22:47.906 substitution. 00:22:49.390 --> 00:22:54.785 If we let X be tongue theater, the integrals going to become 1 00:22:54.785 --> 00:22:59.350 / 1 plus X squared will become 1 + 10 squared. 00:23:00.480 --> 00:23:04.888 Theater. And we have to take care of the DX in an appropriate 00:23:04.888 --> 00:23:11.736 way. Now remember that DX is going to be given by the XD 00:23:11.736 --> 00:23:14.226 theater multiplied by D theater. 00:23:14.370 --> 00:23:18.060 DXD theater we want to differentiate X is 10 theater 00:23:18.060 --> 00:23:19.536 with respect to theater. 00:23:20.450 --> 00:23:24.820 Now the derivative of tongue theater is the secant squared, 00:23:24.820 --> 00:23:27.879 so we get secret squared Theta D 00:23:27.879 --> 00:23:32.943 theater. So this will allow us to change the DX in here. 00:23:33.600 --> 00:23:40.490 Two, secant squared, Theta D Theta over on the right. 00:23:40.490 --> 00:23:44.550 At this stage I'm going to use the trigonometric identity, 00:23:44.550 --> 00:23:50.234 which says that 1 + 10 squared of an angle is equal to the 00:23:50.234 --> 00:23:54.700 sequence squared of the angle. So In other words, all this 00:23:54.700 --> 00:23:58.760 quantity down here is just the sequence squared of Theta. 00:23:58.780 --> 00:24:04.720 And this is very nice now because this term here will 00:24:04.720 --> 00:24:10.660 cancel out with this term down in the denominator down there, 00:24:10.660 --> 00:24:17.140 and we're left purely with the integral of one with respect to 00:24:17.140 --> 00:24:19.840 theater. Very simple to finish. 00:24:20.520 --> 00:24:24.700 The integral of one with respect to theater is just theater. 00:24:24.710 --> 00:24:26.390 Plus a constant of integration. 00:24:28.050 --> 00:24:32.910 We want to return to our original variables and if X was 00:24:32.910 --> 00:24:37.770 10 theater than theater is the angle whose tangent, his ex. So 00:24:37.770 --> 00:24:40.605 theater is 10 to the minus one 00:24:40.605 --> 00:24:43.749 of X. Plus a constant. 00:24:46.010 --> 00:24:47.770 And that's the problem finished. 00:24:48.290 --> 00:24:50.963 This is a very important standard result that the 00:24:50.963 --> 00:24:54.824 integral of one over 1 plus X squared DX is equal to the 00:24:54.824 --> 00:24:58.388 inverse tan 10 to the minus one of X plus a constant. 00:24:58.388 --> 00:25:01.358 That's a result that you'll see in all the standard 00:25:01.358 --> 00:25:04.031 tables of integrals, and it's a result that you'll 00:25:04.031 --> 00:25:07.001 need to call appan very frequently, and if you can't 00:25:07.001 --> 00:25:09.971 remember it, then at least you'll need to know that 00:25:09.971 --> 00:25:13.535 there is such a formula that exists and you want to be 00:25:13.535 --> 00:25:15.020 able to look it up. 00:25:16.720 --> 00:25:20.490 I want to generalize this a little bit to look at the case 00:25:20.490 --> 00:25:24.840 when we deal with not just a one here, but a more general case of 00:25:24.840 --> 00:25:28.320 an arbitrary constant in there. So let's look at what happens if 00:25:28.320 --> 00:25:30.060 we have a situation like this. 00:25:30.900 --> 00:25:36.900 Suppose we want to integrate one over a squared plus X squared 00:25:36.900 --> 00:25:38.900 with respect to X. 00:25:39.480 --> 00:25:42.792 Where a is a 00:25:42.792 --> 00:25:49.857 constant. This time I'm going to make this substitution let X be 00:25:49.857 --> 00:25:55.544 a town theater, and we'll see why we've made that substitution 00:25:55.544 --> 00:25:58.129 in just a little while. 00:25:58.810 --> 00:26:04.410 With this substitution, X is a Tan Theta. The differential 00:26:04.410 --> 00:26:08.890 DX becomes a secant squared Theta D Theta. 00:26:11.690 --> 00:26:14.480 Let's put all this into this 00:26:14.480 --> 00:26:19.705 integral here. Will have the integral of one over a squared. 00:26:20.980 --> 00:26:26.978 Plus And X squared will become a squared 10. 00:26:26.978 --> 00:26:28.451 Squared feet are. 00:26:29.460 --> 00:26:31.805 The 00:26:31.805 --> 00:26:39.434 DX Will become a sex squared Theta D 00:26:39.434 --> 00:26:47.256 Theta. Now what I can do now is I can take out a common 00:26:47.256 --> 00:26:50.208 factor of A squared from the 00:26:50.208 --> 00:26:57.311 denominator. Taking an A squared out from this term will leave me 00:26:57.311 --> 00:27:03.803 one taking a squared out from this term will leave me tan 00:27:03.803 --> 00:27:09.370 squared theater. And it's still on the top. I've got a sex 00:27:09.370 --> 00:27:10.950 squared Theta D Theta. 00:27:13.360 --> 00:27:20.500 We have the trig identity that 1 + 10 squared of any angle is sex 00:27:20.500 --> 00:27:22.404 squared of the angle. 00:27:22.660 --> 00:27:28.861 So I can use that identity in here to write the denominator as 00:27:28.861 --> 00:27:34.585 one over a squared and the 1 + 10 squared becomes simply 00:27:34.585 --> 00:27:36.016 sequence squared theater. 00:27:36.630 --> 00:27:41.442 We still gotten a secant squared theater in the numerator, and a 00:27:41.442 --> 00:27:45.452 lot of this is going to simplify and cancel now. 00:27:46.200 --> 00:27:47.652 The secant squared will go the 00:27:47.652 --> 00:27:52.180 top and the bottom. The one of these at the bottom will go with 00:27:52.180 --> 00:27:56.028 the others at the top, and we're left with the integral of one 00:27:56.028 --> 00:27:57.804 over A with respect to theater. 00:28:00.170 --> 00:28:02.890 Again, this is straightforward to finish. The integral of one 00:28:02.890 --> 00:28:06.426 over a one over as a constant with respect to Theta is just 00:28:06.426 --> 00:28:08.330 going to give me one over a. 00:28:08.870 --> 00:28:11.784 Theater. Plus the constant of 00:28:11.784 --> 00:28:16.846 integration. To return to the original variables, we've got to 00:28:16.846 --> 00:28:21.730 go back to our original substitution. If X is a tan 00:28:21.730 --> 00:28:27.058 Theta, then we can write that X over A is 10 theater. 00:28:27.090 --> 00:28:30.660 And In other words, that theater is the angle whose 00:28:30.660 --> 00:28:35.301 tangent is 10 to the minus one of all this X over a. 00:28:36.590 --> 00:28:41.238 That will enable me to write our final results as one over a town 00:28:41.238 --> 00:28:42.566 to the minus one. 00:28:43.250 --> 00:28:45.938 X over a. 00:28:46.060 --> 00:28:47.620 Plus a constant of integration. 00:28:49.030 --> 00:28:52.540 And this is another very important standard result that 00:28:52.540 --> 00:28:56.830 the integral of one over a squared plus X squared with 00:28:56.830 --> 00:29:03.850 respect to X is one over a 10 to the minus one of X over a plus a 00:29:03.850 --> 00:29:07.750 constant, and as before, that's a standard result that you'll 00:29:07.750 --> 00:29:12.430 see frequently in all the tables of integrals, and you'll need to 00:29:12.430 --> 00:29:16.720 call a pawn that in lots of situations when you're required 00:29:16.720 --> 00:29:17.890 to do integration. 00:29:17.940 --> 00:29:23.940 OK, so now we've got the standard result that the 00:29:23.940 --> 00:29:31.140 integral of one over a squared plus X squared DX is equal 00:29:31.140 --> 00:29:38.340 to one over a town to the minus one of X of 00:29:38.340 --> 00:29:40.400 A. As a constant of integration. 00:29:41.040 --> 00:29:46.408 Let's see how we might use this formula in a slightly 00:29:46.408 --> 00:29:52.264 different case. Suppose we have the integral of 1 / 4 + 00:29:52.264 --> 00:29:54.216 9 X squared DX. 00:29:55.360 --> 00:29:58.517 Now this looks very similar to the standard formula we have 00:29:58.517 --> 00:30:00.770 here. Except there's a slight 00:30:00.770 --> 00:30:04.935 problem. And the problem is that instead of One X squared, which 00:30:04.935 --> 00:30:08.070 we have in the standard result, I've got nine X squared. 00:30:08.850 --> 00:30:11.826 What I'm going to do is I'm going to divide everything at 00:30:11.826 --> 00:30:15.546 the bottom by 9, take a factor of nine out so that we end up 00:30:15.546 --> 00:30:19.266 with just a One X squared here. So what I'm going to do is I'm 00:30:19.266 --> 00:30:20.506 going to write the denominator 00:30:20.506 --> 00:30:25.490 like this. So I've taken a factor of nine out. You'll see 00:30:25.490 --> 00:30:30.050 if we multiply the brackets again here, there's 9 * 4 over 00:30:30.050 --> 00:30:35.370 9, which is just four and the nine times the X squared, so I 00:30:35.370 --> 00:30:39.550 haven't changed anything. I've just taken a factor of nine out 00:30:39.550 --> 00:30:45.250 the point of doing that is that now I have a single. I have a 00:30:45.250 --> 00:30:49.810 One X squared here, which will match the formula I have there. 00:30:50.450 --> 00:30:53.544 If I take the 9 outside the 00:30:53.544 --> 00:30:59.306 integral. I'm left with 1 /, 4 ninths plus X squared integrated 00:30:59.306 --> 00:31:05.546 with respect to X and I hope you can see that this is exactly one 00:31:05.546 --> 00:31:10.954 of the standard forms. Now when we let A squared B4 over nine 00:31:10.954 --> 00:31:16.778 with a squared is 4 over 9. We have the standard form. If A 00:31:16.778 --> 00:31:23.018 squared is 4 over 9A will be 2 over 3 and we can complete this 00:31:23.018 --> 00:31:27.527 integration. Using the standard result that one over 9 stays 00:31:27.527 --> 00:31:29.765 there, we want one over A. 00:31:30.540 --> 00:31:34.740 Or A is 2/3. So we want 1 / 2/3. 00:31:35.810 --> 00:31:37.650 10 to the minus one. 00:31:38.390 --> 00:31:40.238 Of X over a. 00:31:40.880 --> 00:31:44.800 X divided by a is X divided by 00:31:44.800 --> 00:31:48.238 2/3. Plus a constant of 00:31:48.238 --> 00:31:53.453 integration. Just to tide to these fractions up, three will 00:31:53.453 --> 00:31:58.040 divide into 9 three times, so we'll have 326 in the 00:31:58.040 --> 00:32:03.708 denominator. 10 to the minus one and dividing by 2/3 is like 00:32:03.708 --> 00:32:09.560 multiplying by three over 2, so I'll have 10 to the minus one of 00:32:09.560 --> 00:32:12.486 three X over 2 plus the constant 00:32:12.486 --> 00:32:16.549 of integration. So the point here is you might have to do a 00:32:16.549 --> 00:32:19.396 bit of work on the integrand in order to be able to write 00:32:19.396 --> 00:32:21.586 it in the form of one of the standard results. 00:32:22.870 --> 00:32:28.870 OK, let's have a look at another case where another integral to 00:32:28.870 --> 00:32:32.870 look at where a trigonometric substitution is appropriate. 00:32:32.870 --> 00:32:38.870 Suppose we want to find the integral of one over the square 00:32:38.870 --> 00:32:41.870 root of A squared minus X 00:32:41.870 --> 00:32:47.160 squared DX. Again, A is a constant. 00:32:49.710 --> 00:32:55.610 The substitution that I'm going to make is this one. 00:32:55.610 --> 00:33:00.920 I'm going to write X equals a sign theater. 00:33:02.130 --> 00:33:08.550 If I do that, what will happen to my integral, let's see. 00:33:09.080 --> 00:33:11.162 And have the integral of one 00:33:11.162 --> 00:33:17.976 over. The square root. The A squared will stay the same, but 00:33:17.976 --> 00:33:22.800 the X squared will become a squared sine squared. 00:33:22.800 --> 00:33:25.970 I squared sine squared Theta. 00:33:26.750 --> 00:33:30.794 Now the reason I've done that is because in a minute I'm 00:33:30.794 --> 00:33:35.175 going to take out a factor of a squared, which will leave me 00:33:35.175 --> 00:33:39.556 one 1 minus sign squared, and I do have an identity involving 1 00:33:39.556 --> 00:33:43.263 minus sign squared as we'll see, but just before we do 00:33:43.263 --> 00:33:47.307 that, let's substitute for the differential as well. If X is a 00:33:47.307 --> 00:33:51.014 sign theater, then DX will be a cosine, Theta, D, Theta. 00:33:52.350 --> 00:33:59.034 So we have a cosine Theta D Theta for the differential DX. 00:34:01.400 --> 00:34:07.388 Let me take out the factor of a squared in the denominator. 00:34:08.040 --> 00:34:13.837 Taking a squad from this first term will leave me one 00:34:13.837 --> 00:34:20.161 and a squared from the second term will leave me one minus 00:34:20.161 --> 00:34:21.742 sign squared Theta. 00:34:22.810 --> 00:34:27.094 I have still gotten a costly to the theater at the top. 00:34:28.490 --> 00:34:31.934 Now let me remind you there's a trig identity which says that 00:34:31.934 --> 00:34:35.378 the cosine squared of an angle plus the sine squared of an 00:34:35.378 --> 00:34:36.526 angle is always one. 00:34:37.160 --> 00:34:40.758 So if we have one minus the sine squared of an angle, we can 00:34:40.758 --> 00:34:42.043 replace it with cosine squared. 00:34:42.670 --> 00:34:49.633 So 1 minus sign squared Theta we can replace with simply 00:34:49.633 --> 00:34:51.532 cosine squared Theta. 00:34:51.540 --> 00:34:54.408 Is the A squared out the frontier and we want the square 00:34:54.408 --> 00:34:55.603 root of the whole lot. 00:34:56.280 --> 00:35:03.352 Now this is very simple. We want the square root of A squared 00:35:03.352 --> 00:35:08.792 cosine squared Theta. We square root. These squared terms will 00:35:08.792 --> 00:35:10.968 be just left with. 00:35:10.980 --> 00:35:12.600 A cosine Theta. 00:35:13.140 --> 00:35:18.618 In the denominator and within a cosine Theta in the numerator. 00:35:19.770 --> 00:35:21.960 And these were clearly cancel out. 00:35:23.080 --> 00:35:27.700 And we're left with the integral of one with respect to theater, 00:35:27.700 --> 00:35:31.165 which is just theater plus a constant of integration. 00:35:33.940 --> 00:35:39.232 Just to return to the original variables, given that X was a 00:35:39.232 --> 00:35:43.642 sign theater, then clearly X over A is sign theater. 00:35:44.400 --> 00:35:50.048 So theater is the angle who sign is or sign to the minus one of X 00:35:50.048 --> 00:35:54.990 over a, so replacing the theater with sign to the minus one of X 00:35:54.990 --> 00:35:57.108 over a will get this result. 00:35:57.650 --> 00:36:02.402 And this is a very important standard result that if you want 00:36:02.402 --> 00:36:07.550 to integrate 1 divided by the square root of A squared minus X 00:36:07.550 --> 00:36:12.302 squared, the result is the inverse sine or the sign to the 00:36:12.302 --> 00:36:14.678 minus one of X over a. 00:36:15.250 --> 00:36:16.670 Plus a constant of integration. 00:36:18.010 --> 00:36:25.258 Will have a look one final example which is a variant on 00:36:25.258 --> 00:36:31.902 the previous example. Suppose we want to integrate 1 divided by 00:36:31.902 --> 00:36:37.942 the square root of 4 - 9 X squared DX. 00:36:38.570 --> 00:36:42.902 Now that's very similar to the one we just looked at. Remember 00:36:42.902 --> 00:36:47.595 that we had the results that the integral of one over the square 00:36:47.595 --> 00:36:49.761 root of A squared minus X 00:36:49.761 --> 00:36:55.381 squared DX. Was the inverse sine of X over a plus a constant? 00:36:55.381 --> 00:36:59.291 That's keep that in mind. That's the standard result we've 00:36:59.291 --> 00:37:03.860 already proved. We're almost there. In this case. The problem 00:37:03.860 --> 00:37:08.120 is that instead of a single X squared, we've got nine X 00:37:08.120 --> 00:37:12.192 squared. So like we did in the other example, I'm going to take 00:37:12.192 --> 00:37:15.776 the factor of nine out to leave us just a single X squared in 00:37:15.776 --> 00:37:17.568 there, and I do that like this. 00:37:18.840 --> 00:37:26.060 Taking a nine out from these terms here, I'll have 00:37:26.060 --> 00:37:29.670 four ninths minus X squared. 00:37:30.290 --> 00:37:33.746 Again, the nine times the four ninths leaves the four which we 00:37:33.746 --> 00:37:37.202 had originally, and then we've got the nine X squared, which we 00:37:37.202 --> 00:37:41.910 have there. The whole point of doing that is that then I'm 00:37:41.910 --> 00:37:46.187 going to extract the Route 9, which is 3 and bring it right 00:37:46.187 --> 00:37:52.290 outside. And inside under the integral sign, I'll be left with 00:37:52.290 --> 00:37:58.314 one over the square root of 4 ninths minus X squared DX. 00:38:00.340 --> 00:38:05.464 Now in this form, I hope you can spot that we can use the 00:38:05.464 --> 00:38:08.758 standard result immediately with the standard results, with a 00:38:08.758 --> 00:38:12.052 being with a squared being equal to four ninths. 00:38:12.790 --> 00:38:16.696 In other words, a being equal to 00:38:16.696 --> 00:38:21.140 2/3. Putting all that together will have a third. That's the 00:38:21.140 --> 00:38:24.632 third and the integral will become the inverse sine. 00:38:25.940 --> 00:38:29.674 X. Divided by AA 00:38:29.674 --> 00:38:35.170 was 2/3. Plus a constant of integration. 00:38:37.140 --> 00:38:43.006 And just to tidy that up will be left with the third inverse sine 00:38:43.006 --> 00:38:48.453 dividing by 2/3 is the same as multiplying by three over 2, so 00:38:48.453 --> 00:38:52.643 will have 3X over 2 plus a constant of integration. 00:38:52.670 --> 00:38:56.060 And that's our final result. So we've seen a lot 00:38:56.060 --> 00:38:58.094 of examples that have integration using 00:38:58.094 --> 00:39:00.128 trigonometric identities and integration using trig 00:39:00.128 --> 00:39:03.179 substitutions. You need a lot of practice, and there 00:39:03.179 --> 00:39:06.230 are a lot of exercises in the accompanying text.