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