1 00:00:01,340 --> 00:00:04,616 Sometimes integrals involving trigonometric functions can be 2 00:00:04,616 --> 00:00:08,828 evaluated by first of all using trigonometric identities to 3 00:00:08,828 --> 00:00:13,040 rewrite the integrand. That's the quantity we're trying to 4 00:00:13,040 --> 00:00:18,188 integrate an alternative form, which is a bit more amenable to 5 00:00:18,188 --> 00:00:21,645 integration. Sometimes a trigonometric substitution is 6 00:00:21,645 --> 00:00:26,175 appropriate. Both of these techniques we look at in this 7 00:00:26,175 --> 00:00:31,034 unit. Before we start I want to give you a couple of preliminary 8 00:00:31,034 --> 00:00:35,025 results which will be using over and over again and which will be 9 00:00:35,025 --> 00:00:39,323 very important and the first one is I want you to make sure that 10 00:00:39,323 --> 00:00:43,007 you know that the integral of the cosine of a constant times 11 00:00:43,007 --> 00:00:45,768 X. With respect to X. 12 00:00:46,290 --> 00:00:49,699 Is equal to one over that constant. 13 00:00:50,850 --> 00:00:55,228 Multiplied by the sign of KX plus a constant of integration 14 00:00:55,228 --> 00:00:59,606 as a very important result. If you integrate the cosine, you 15 00:00:59,606 --> 00:01:00,800 get a sign. 16 00:01:01,970 --> 00:01:05,415 And if there's a constant in front of the X that appears down 17 00:01:05,415 --> 00:01:08,595 here will take that as read in all the examples which follow. 18 00:01:09,360 --> 00:01:15,245 Another important results is the integral of a sign. The integral 19 00:01:15,245 --> 00:01:22,735 of sine KX with respect to X is minus one over K cosine KX. 20 00:01:23,390 --> 00:01:26,950 Plus a constant we're integrating assign. The result 21 00:01:26,950 --> 00:01:31,400 is minus the cosine and the constant factor. There appears 22 00:01:31,400 --> 00:01:34,515 out down here as well, so those 23 00:01:34,515 --> 00:01:37,300 two results. Very important. 24 00:01:37,900 --> 00:01:40,720 You should have them at your fingertips and we can call upon 25 00:01:40,720 --> 00:01:43,305 them whenever we want them in the rest of the video. 26 00:01:44,090 --> 00:01:47,897 We also want to call appan trigonometric identity's. I'm 27 00:01:47,897 --> 00:01:52,127 going to assume that you've seen a lot of trigonometric 28 00:01:52,127 --> 00:01:55,382 identities before. We have a table of trigonometric 29 00:01:55,382 --> 00:01:58,649 identities here, such as the table that you might have seen 30 00:01:58,649 --> 00:02:01,916 many times before. If you want this specific table, you'll find 31 00:02:01,916 --> 00:02:03,401 it in the printed notes 32 00:02:03,401 --> 00:02:07,022 accompanying the video. Why might we want to use 33 00:02:07,022 --> 00:02:09,980 trigonometric identities? Well, for example, we've just 34 00:02:09,980 --> 00:02:14,108 seen that we already know how to integrate the sign of a 35 00:02:14,108 --> 00:02:17,892 quantity and the cosine of the quantity. But suppose we want 36 00:02:17,892 --> 00:02:21,332 to integrate assign multiplied by a cosine or cosine times 37 00:02:21,332 --> 00:02:24,428 cosine or assigned times assign. We don't actually know 38 00:02:24,428 --> 00:02:27,868 how to do those integrals. Integrals at the moment, but 39 00:02:27,868 --> 00:02:30,620 if we use trigonometric identities, we can rewrite 40 00:02:30,620 --> 00:02:34,404 these in terms of just single sine and cosine terms, which 41 00:02:34,404 --> 00:02:35,780 we can then integrate. 42 00:02:36,900 --> 00:02:39,665 Also, the trigonometric identities identities allow us 43 00:02:39,665 --> 00:02:44,010 to integrate powers of sines and cosines. You'll see that using 44 00:02:44,010 --> 00:02:47,960 these identity's? We've got powers of cosine powers of sign 45 00:02:47,960 --> 00:02:53,095 and the identity is allow us to write into grams in terms of 46 00:02:53,095 --> 00:02:55,070 cosines and sines of double 47 00:02:55,070 --> 00:02:59,070 angles. We know how to integrate these already using the results. 48 00:02:59,070 --> 00:03:02,646 I've just reminded you of, so I'm going to assume that you've 49 00:03:02,646 --> 00:03:06,222 got a table like this at your fingertips, and we can call 50 00:03:06,222 --> 00:03:08,010 appan it whenever we need to. 51 00:03:08,500 --> 00:03:12,472 OK, let's have a look at the first example and the example 52 00:03:12,472 --> 00:03:16,444 that I'm going to look at is a definite integral. The integral 53 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 54 00:03:22,402 --> 00:03:25,712 particular, we've gotta power here. We're looking at the sign 55 00:03:25,712 --> 00:03:30,346 squared of X. What I'm going to do is go back to the table. 56 00:03:31,070 --> 00:03:34,931 And look for an identity that will allow us to change the sign 57 00:03:34,931 --> 00:03:38,495 squared X into something else. Let me just flip back to the 58 00:03:38,495 --> 00:03:39,386 table of trigonometric 59 00:03:39,386 --> 00:03:43,960 identities. The identity that I'm going to use this one, the 60 00:03:43,960 --> 00:03:45,079 cosine of 2A. 61 00:03:45,650 --> 00:03:48,618 Is 1 minus twice sign square day? 62 00:03:49,650 --> 00:03:52,884 If you inspect this carefully, you'll see that this will enable 63 00:03:52,884 --> 00:03:56,706 us to change a sine squared into the cosine of a double angle. 64 00:03:57,690 --> 00:04:00,576 Let me write that down again. 65 00:04:00,580 --> 00:04:07,396 Cosine of 2 A is equal to 1 66 00:04:07,396 --> 00:04:10,804 minus twice sign squared 67 00:04:10,804 --> 00:04:13,990 A. First of all, I'm going to rearrange this to get 68 00:04:13,990 --> 00:04:15,155 sine squared on its own. 69 00:04:16,260 --> 00:04:20,730 If we add two sine squared data both sides, then I can get it on 70 00:04:20,730 --> 00:04:27,060 this side. And if I subtract cosine 2A from both sides, are 71 00:04:27,060 --> 00:04:29,665 remove it from the left. 72 00:04:29,790 --> 00:04:37,190 Finally, if I divide both sides by two, I'll be 73 00:04:37,190 --> 00:04:40,890 left with sine squared A. 74 00:04:40,900 --> 00:04:46,640 And this is the result that I want to use to help me to 75 00:04:46,640 --> 00:04:51,560 evaluate this integral because of what it will allow me to do. 76 00:04:51,560 --> 00:04:56,890 Is it will allow me to change a quantity involving the square of 77 00:04:56,890 --> 00:05:00,990 a trig function into a quantity involving double angles. So 78 00:05:00,990 --> 00:05:03,450 let's use it in this case. 79 00:05:04,410 --> 00:05:10,470 The integral will become the integral from note to pie. 80 00:05:11,340 --> 00:05:18,446 Sine squared X using this formula will be 1 minus cosine 81 00:05:18,446 --> 00:05:22,266 twice X. All divided by 82 00:05:22,266 --> 00:05:24,710 two. Integrated with respect to X. 83 00:05:26,120 --> 00:05:31,632 I've taken out the fact that 1/2 here and I'm left with the 84 00:05:31,632 --> 00:05:36,720 numerator 1 minus cosine 2X to be integrated with respect to X. 85 00:05:37,620 --> 00:05:39,120 This is straightforward to 86 00:05:39,120 --> 00:05:43,884 finish off. So definite integral. So I have square 87 00:05:43,884 --> 00:05:49,452 brackets. The integral of one with respect to X is simply X. 88 00:05:50,470 --> 00:05:55,030 And the integral of cosine 2 X we know from our preliminary 89 00:05:55,030 --> 00:06:00,350 work is just going to be sine 2X divided by two with a minus 90 00:06:00,350 --> 00:06:03,770 sign there and the limits are not and pie. 91 00:06:05,770 --> 00:06:10,424 We finish this off by first of all, putting the upper limit in, 92 00:06:10,424 --> 00:06:14,362 so we want X replaced by pie here and pie here. 93 00:06:15,070 --> 00:06:17,620 The sign of 2π is 0. 94 00:06:18,520 --> 00:06:22,645 So when we put the upper limit in will just get. 95 00:06:22,930 --> 00:06:26,320 Pie by substituting for X here. 96 00:06:27,030 --> 00:06:30,880 Let me put the lower limit in. 97 00:06:30,880 --> 00:06:33,547 X being not will be 0 here. 98 00:06:34,210 --> 00:06:38,830 And sign of note here, which is not so both of those terms will 99 00:06:38,830 --> 00:06:43,120 become zero when we put the lower limit in and so we're just 100 00:06:43,120 --> 00:06:45,760 left with simply 1/2 of Π or π 101 00:06:45,760 --> 00:06:51,040 by 2. And that's our first example of how we've used a 102 00:06:51,040 --> 00:06:53,686 trigonometric identity to rewrite an integrand involving 103 00:06:53,686 --> 00:06:58,222 powers of a trig function in terms of double angles, which we 104 00:06:58,222 --> 00:07:00,112 already know how to integrate. 105 00:07:01,360 --> 00:07:09,190 Let's have a look at another example. Suppose we want 106 00:07:09,190 --> 00:07:17,020 to integrate the sign of three X multiplied by the 107 00:07:17,020 --> 00:07:20,152 cosine of 2 X. 108 00:07:20,160 --> 00:07:21,248 With respect to X. 109 00:07:21,770 --> 00:07:26,306 Now we already know how to integrate signs. We know how to 110 00:07:26,306 --> 00:07:30,464 integrate cosines, but we have a problem here because there's a 111 00:07:30,464 --> 00:07:34,244 product. These two terms are multiplied together and we don't 112 00:07:34,244 --> 00:07:35,756 know how to proceed. 113 00:07:36,590 --> 00:07:41,282 What we do is look in our table of trigonometric identities for 114 00:07:41,282 --> 00:07:45,192 an example where we've gotta sign multiplied by a cosine. 115 00:07:45,192 --> 00:07:47,538 Let's go back to the table. 116 00:07:47,560 --> 00:07:53,743 The first entry in our table involves assign multiplied 117 00:07:53,743 --> 00:07:55,804 by a cosine. 118 00:07:56,660 --> 00:08:02,048 Let me write this formula down again. 2 sign a cosine be. 119 00:08:02,050 --> 00:08:07,690 Is equal 120 00:08:07,690 --> 00:08:14,302 to. The sign of the sum of A and be 121 00:08:14,302 --> 00:08:16,450 added to the sign of the 122 00:08:16,450 --> 00:08:23,935 difference A-B. And this is the identity that I 123 00:08:23,935 --> 00:08:31,655 will use in order to rewrite this integrand 124 00:08:31,655 --> 00:08:39,375 as two separate integrals. We identify the A's 125 00:08:39,375 --> 00:08:43,280 3X. The B is 2 X. 126 00:08:44,050 --> 00:08:48,714 The factor of 2 here isn't a problem. We can divide 127 00:08:48,714 --> 00:08:50,410 everything through by two. 128 00:08:50,420 --> 00:08:51,785 So we lose it from this side. 129 00:08:52,770 --> 00:08:57,973 So our integral? What will it become? Well, the integral of 130 00:08:57,973 --> 00:09:03,649 sign 3X cosine 2X DX will become. We want the integral of 131 00:09:03,649 --> 00:09:06,960 the sign of the sum of A&B. 132 00:09:07,540 --> 00:09:12,244 Well, there's some of A&B will be 3X plus 2X, which is 5X. So 133 00:09:12,244 --> 00:09:14,260 we want the sign of 5X. 134 00:09:15,520 --> 00:09:19,502 Added to the sign of the difference of amb. Well a 135 00:09:19,502 --> 00:09:23,846 being 3X B being 2X A-B will be 3X subtract 2 X 136 00:09:23,846 --> 00:09:28,552 which is just One X. So we want the sign of X all 137 00:09:28,552 --> 00:09:32,172 divided by two and we want to integrate that with 138 00:09:32,172 --> 00:09:33,258 respect to X. 139 00:09:34,510 --> 00:09:38,995 So what have we done? We've used the trig identity to change the 140 00:09:38,995 --> 00:09:43,135 product of a signing cosine into the sum of two separate sign 141 00:09:43,135 --> 00:09:46,585 terms, which we can integrate straight away. We can integrate 142 00:09:46,585 --> 00:09:48,655 that taking the factor of 1/2 143 00:09:48,655 --> 00:09:56,314 out. The integral of sign 5X will be minus the cosine of 5X 144 00:09:56,314 --> 00:09:58,000 divided by 5. 145 00:09:58,740 --> 00:10:03,628 And the integral of sine X will be just minus cosine X, and 146 00:10:03,628 --> 00:10:05,508 they'll be a constant of 147 00:10:05,508 --> 00:10:10,924 integration. And just to tidy it up, at the end we're going to 148 00:10:10,924 --> 00:10:14,968 have minus the half with the five at the bottom. There will 149 00:10:14,968 --> 00:10:18,001 give you minus cosine 5X all divided by 10. 150 00:10:19,060 --> 00:10:23,092 And there's a half with this term here, so it's minus cosine 151 00:10:23,092 --> 00:10:24,436 X divided by two. 152 00:10:25,220 --> 00:10:27,950 Plus a constant of integration. 153 00:10:28,500 --> 00:10:30,288 And that's the solution of this 154 00:10:30,288 --> 00:10:35,420 problem. Let's explore the integral of products of sines 155 00:10:35,420 --> 00:10:41,900 and cosines a little bit further, and what I want to look 156 00:10:41,900 --> 00:10:48,920 at now is integrals of the form the integral of sign to the 157 00:10:48,920 --> 00:10:54,320 power MX multiplied by cosine to the power NX DX. 158 00:10:54,930 --> 00:10:58,230 Well, look at a whole family of integrals like this, but in 159 00:10:58,230 --> 00:11:01,805 particular for the first example I'm going to look at the case of 160 00:11:01,805 --> 00:11:03,730 what happens when M is an odd 161 00:11:03,730 --> 00:11:09,300 number. Whenever you have an integral like this, when M is 162 00:11:09,300 --> 00:11:14,239 odd, the following process will work. Let's look at a specific 163 00:11:14,239 --> 00:11:18,280 case, supposing I want to integrate sine cubed X. 164 00:11:18,820 --> 00:11:23,500 Multiplied by cosine squared XDX. 165 00:11:24,590 --> 00:11:27,530 Notice that M. 166 00:11:28,220 --> 00:11:30,670 Is an odd number and is 3. 167 00:11:31,560 --> 00:11:35,356 There's a little trick here that we're going to do now, and it's 168 00:11:35,356 --> 00:11:38,860 the sort of trick that comes with practice and seeing lots of 169 00:11:38,860 --> 00:11:42,364 examples. What we're going to do is we're going to rewrite the 170 00:11:42,364 --> 00:11:44,116 sign cubed X in a slightly 171 00:11:44,116 --> 00:11:49,285 different form. We're going to recognize that sign cubed can be 172 00:11:49,285 --> 00:11:53,290 written as sine squared X multiplied by Sign X. 173 00:11:53,800 --> 00:11:57,568 That's a little trick. The sign cubed can be written as sine 174 00:11:57,568 --> 00:12:01,410 squared times sign. So our integral can be 175 00:12:01,410 --> 00:12:05,202 written as sine squared X times sign X 176 00:12:05,202 --> 00:12:08,046 multiplied by cosine squared X DX. 177 00:12:09,240 --> 00:12:12,670 And then I'm going to pick a trigonometric identity involving 178 00:12:12,670 --> 00:12:16,443 sine squared to write it in terms of cosine squared. Let's 179 00:12:16,443 --> 00:12:17,472 find that identity. 180 00:12:18,090 --> 00:12:21,115 With an identity here, which says that sine squared of an 181 00:12:21,115 --> 00:12:22,765 angle plus cost squared of an 182 00:12:22,765 --> 00:12:27,834 angle is one. If we rearrange this, we can write that sine 183 00:12:27,834 --> 00:12:32,722 squared of an angle is 1 minus the cosine squared of an angle 184 00:12:32,722 --> 00:12:33,850 will use that. 185 00:12:34,700 --> 00:12:38,596 Sine squared of any 186 00:12:38,596 --> 00:12:44,860 angle. Is equal to 1 minus the cosine squared over any 187 00:12:44,860 --> 00:12:51,290 angle. Will use that in here to change the sign squared X into 188 00:12:51,290 --> 00:12:55,740 terms involving cosine squared X. Let's see what happens. This 189 00:12:55,740 --> 00:13:00,190 integral will become the integral of or sign squared X. 190 00:13:00,790 --> 00:13:03,500 Will become one minus cosine 191 00:13:03,500 --> 00:13:09,320 squared X. There's still the terms cynex. 192 00:13:11,620 --> 00:13:13,657 And at the end we still got 193 00:13:13,657 --> 00:13:17,378 cosine squared X. Now this is looking a bit complicated, but 194 00:13:17,378 --> 00:13:20,641 as we'll see it's all going to come out in the Wash. Let's 195 00:13:20,641 --> 00:13:22,147 remove the brackets here and see 196 00:13:22,147 --> 00:13:27,251 what we've got. There's a one multiplied by all this sign X 197 00:13:27,251 --> 00:13:28,767 times cosine squared X. 198 00:13:29,440 --> 00:13:33,265 So that's just sign X times cosine squared X 199 00:13:33,265 --> 00:13:37,090 will want to integrate that with respect to X. 200 00:13:38,510 --> 00:13:42,443 There's also cosine squared X multiplied by all this. 201 00:13:42,980 --> 00:13:47,468 Now the cosine squared X with this cosine squared X will give 202 00:13:47,468 --> 00:13:50,086 us a cosine, so the power 4X. 203 00:13:51,840 --> 00:13:53,920 There's also the sign X. 204 00:13:54,850 --> 00:13:56,800 And we want to integrate that. 205 00:13:57,380 --> 00:14:00,851 Also, with respect to X and there was a minus sign in front, 206 00:14:00,851 --> 00:14:02,720 so that's going to go in there. 207 00:14:03,350 --> 00:14:05,702 So we've expanded the brackets here and written. 208 00:14:05,702 --> 00:14:07,172 This is 2 separate integrals. 209 00:14:08,410 --> 00:14:13,591 Now, each of these integrals can be evaluated by making a 210 00:14:13,591 --> 00:14:18,301 substitution. If we make a substitution and let you equals 211 00:14:18,301 --> 00:14:24,168 cosine X. The differential du is du DX. 212 00:14:24,970 --> 00:14:30,880 DX Do you DX if we differentiate cosine, X will get 213 00:14:30,880 --> 00:14:32,564 minus the sign X. 214 00:14:33,110 --> 00:14:36,582 So we've got du is minus sign X 215 00:14:36,582 --> 00:14:42,560 DX. Now look at what we've got when we make this substitution. 216 00:14:42,560 --> 00:14:48,020 The cosine squared X will become simply you squared and sign X DX 217 00:14:48,020 --> 00:14:53,060 altogether can be written as a minus du, so this will become. 218 00:14:53,830 --> 00:14:55,318 Minus the integral. 219 00:14:56,010 --> 00:14:57,420 Of you squared. 220 00:14:58,010 --> 00:14:58,940 Do you? 221 00:15:01,250 --> 00:15:06,398 What about this term? We've got cosine to the power four cosine 222 00:15:06,398 --> 00:15:09,830 to the power 4X will be you to 223 00:15:09,830 --> 00:15:15,240 the powerful. And sign X DX sign X DX is minus DU. 224 00:15:15,240 --> 00:15:18,425 There's another minus sign here, so overall 225 00:15:18,425 --> 00:15:22,520 will have plus the integral of you to the 226 00:15:22,520 --> 00:15:23,885 four, do you? 227 00:15:25,450 --> 00:15:29,641 Now these are very very simple integrals to finish the integral 228 00:15:29,641 --> 00:15:32,308 of you squared is you cubed over 229 00:15:32,308 --> 00:15:38,630 3? The integral of you to the four is due to the five over 5 230 00:15:38,630 --> 00:15:40,455 plus a constant of integration. 231 00:15:42,470 --> 00:15:48,008 All we need to do to finish off is return to our original 232 00:15:48,008 --> 00:15:53,120 variables. Remember, you was cosine of X, so we finish off by 233 00:15:53,120 --> 00:15:54,398 writing minus 1/3. 234 00:15:54,970 --> 00:15:59,304 You being cosine X means that we've got cosine cubed X. 235 00:16:00,670 --> 00:16:07,180 Plus 1/5. You to the five will be Co sign 236 00:16:07,180 --> 00:16:09,100 to the power 5X. 237 00:16:10,260 --> 00:16:11,760 Plus a constant of integration. 238 00:16:12,570 --> 00:16:17,190 And that's the solution to the problem that we started with. 239 00:16:18,220 --> 00:16:24,184 Let's stick with the same sort of family of integrals, so we're 240 00:16:24,184 --> 00:16:30,148 still sticking with the integral of sign to the power MX cosine 241 00:16:30,148 --> 00:16:32,633 to the power NX DX. 242 00:16:33,210 --> 00:16:37,650 And now I'm going to have a look at what happens in the case when 243 00:16:37,650 --> 00:16:39,130 M is an even number. 244 00:16:39,650 --> 00:16:42,560 And N is an odd number. 245 00:16:44,480 --> 00:16:47,252 This method will always work when M is even. An is odd. 246 00:16:47,790 --> 00:16:52,266 Let's look at a specific case. Suppose we want to integrate the 247 00:16:52,266 --> 00:16:54,131 sign to the power 4X. 248 00:16:55,190 --> 00:16:57,749 Cosine cubed X. 249 00:16:58,350 --> 00:16:59,290 DX 250 00:17:01,840 --> 00:17:07,118 Notice that M the power of sign is now even em is full. 251 00:17:08,430 --> 00:17:12,690 And N which is the power of cosine, is odd an IS3. 252 00:17:13,430 --> 00:17:17,343 What I'm going to do is I'm going to use the identity that 253 00:17:17,343 --> 00:17:21,256 cosine squared of an angle is 1 minus sign squared of an angle 254 00:17:21,256 --> 00:17:25,169 and you'll be able to lift that directly from the table we had 255 00:17:25,169 --> 00:17:28,179 at the beginning, which stated the very important and well 256 00:17:28,179 --> 00:17:31,189 known results that cosine squared of an angle plus the 257 00:17:31,189 --> 00:17:34,199 sine squared of an angle is always equal to 1. 258 00:17:34,890 --> 00:17:40,112 What I'm going to do is I'm going to use this to rewrite the 259 00:17:40,112 --> 00:17:42,850 cosine term. In here, in terms 260 00:17:42,850 --> 00:17:47,290 of signs. First of all, I'm going to apply the little trick 261 00:17:47,290 --> 00:17:53,284 we had before. And split the cosine turn up like this cosine 262 00:17:53,284 --> 00:17:56,310 cubed. I'm going to write this 263 00:17:56,310 --> 00:17:59,280 cosine squared. Multiplied by 264 00:17:59,280 --> 00:18:05,400 cosine. So I've changed the cosine cubed to these two terms 265 00:18:05,400 --> 00:18:12,220 here. Now I can use the identity to change cosine 266 00:18:12,220 --> 00:18:15,370 squared X into terms involving 267 00:18:15,370 --> 00:18:20,558 sine squared. So the integral will become the integral of 268 00:18:20,558 --> 00:18:23,248 sign. To the power 4X. 269 00:18:24,060 --> 00:18:30,000 Cosine squared X. We can write as one minus sign, squared X. 270 00:18:31,460 --> 00:18:35,940 And there's still this term cosine X here as well. 271 00:18:37,730 --> 00:18:41,305 And all that has to be integrated with respect to X. 272 00:18:44,240 --> 00:18:48,981 Let me remove the brackets here. When we remove the brackets, 273 00:18:48,981 --> 00:18:54,584 there will be signed to the 4th X Times one all multiplied by 274 00:18:54,584 --> 00:19:01,685 cosine X. That'll be signed to the 4th X 275 00:19:01,685 --> 00:19:05,185 multiplied by sign squared 276 00:19:05,185 --> 00:19:10,676 X. Which is signed to the 6X or multiplied by cosine X. 277 00:19:12,130 --> 00:19:19,020 And there's a minus sign in the middle, and we want to integrate 278 00:19:19,020 --> 00:19:23,150 all that. With respect to X. 279 00:19:25,340 --> 00:19:29,100 Again, a simple substitution will allow us to finish this 280 00:19:29,100 --> 00:19:30,980 off. If we let you. 281 00:19:31,510 --> 00:19:33,250 Be sign X. 282 00:19:34,350 --> 00:19:35,970 So do you. 283 00:19:36,480 --> 00:19:39,140 Is cosine X DX. 284 00:19:39,790 --> 00:19:43,563 This will become immediately the integral of well signed to the 285 00:19:43,563 --> 00:19:48,022 4th X sign to the 4th X will be you to the four. 286 00:19:48,780 --> 00:19:54,126 The cosine X times the DX cosine X DX becomes du. 287 00:19:55,720 --> 00:20:02,520 Subtract. Sign to the six, X will become you to 288 00:20:02,520 --> 00:20:09,386 the six. And the cosine X DX is du. 289 00:20:09,470 --> 00:20:13,199 So what we've achieved are two very simple integrals that we 290 00:20:13,199 --> 00:20:14,894 can complete to finish the 291 00:20:14,894 --> 00:20:20,754 problem. The integral of you to the four is due to the five over 292 00:20:20,754 --> 00:20:25,596 5. The integral of you to the six is due to the 7 over 7. 293 00:20:26,460 --> 00:20:27,549 Plus a constant. 294 00:20:29,010 --> 00:20:33,690 And then just to finish off, we return to the original variables 295 00:20:33,690 --> 00:20:37,980 and replace EU with sign X, which will give us 1/5. 296 00:20:38,510 --> 00:20:42,360 Sign next to the five or sign to the power 5X. 297 00:20:44,160 --> 00:20:45,090 Minus. 298 00:20:46,110 --> 00:20:52,440 One 7th. You to the Seven will be signed to the 7X. 299 00:20:53,050 --> 00:20:56,270 Plus a constant of integration. 300 00:20:58,300 --> 00:21:01,708 So that's how we deal with integrals of this family. In the 301 00:21:01,708 --> 00:21:05,968 case when M is an even number and when N is an odd number. Now 302 00:21:05,968 --> 00:21:09,660 in the case when both M&N are even, you should try using the 303 00:21:09,660 --> 00:21:13,068 double angle formulas, and I'm not going to do an example of 304 00:21:13,068 --> 00:21:16,760 that because there isn't time in this video to do that. But there 305 00:21:16,760 --> 00:21:19,600 are examples in the exercises accompanying the video and you 306 00:21:19,600 --> 00:21:21,020 should try those for yourself. 307 00:21:21,730 --> 00:21:28,610 I'm not going to look at some integrals for which 308 00:21:28,610 --> 00:21:31,362 a trigonometric substitution is 309 00:21:31,362 --> 00:21:36,787 appropriate. Suppose we want to evaluate this integral. 310 00:21:36,790 --> 00:21:43,102 The integral of 1 / 1 311 00:21:43,102 --> 00:21:46,258 plus X squared. 312 00:21:47,030 --> 00:21:48,178 With respect to X. 313 00:21:49,710 --> 00:21:53,103 Now the trigonometric substitution that I want to use 314 00:21:53,103 --> 00:21:59,135 is this one. I want to let X be the tangent of a new variable, X 315 00:21:59,135 --> 00:22:00,266 equals 10 theater. 316 00:22:00,920 --> 00:22:04,115 While I picked this particular substitution well, all will 317 00:22:04,115 --> 00:22:09,085 become clear in time, but I want to just look ahead a little bit 318 00:22:09,085 --> 00:22:11,215 by letting X equal 10 theater. 319 00:22:11,750 --> 00:22:14,837 What will have at the denominator down here is 320 00:22:14,837 --> 00:22:16,552 1 + 10 squared theater. 321 00:22:17,570 --> 00:22:22,946 One plus X squared will become 1 + 10 squared and we have an 322 00:22:22,946 --> 00:22:27,170 identity already which says that 1 + 10 squared of an 323 00:22:27,170 --> 00:22:31,394 angle is equal to the sequence squared of the angle. That's 324 00:22:31,394 --> 00:22:36,002 an identity that we had on the table right at the beginning, 325 00:22:36,002 --> 00:22:40,610 so the idea is that by making this substitution, 1 + 10 326 00:22:40,610 --> 00:22:44,450 squared can be replaced by a single term sequence squared, 327 00:22:44,450 --> 00:22:47,522 as we'll see, so let's progress with that 328 00:22:47,522 --> 00:22:47,906 substitution. 329 00:22:49,390 --> 00:22:54,785 If we let X be tongue theater, the integrals going to become 1 330 00:22:54,785 --> 00:22:59,350 / 1 plus X squared will become 1 + 10 squared. 331 00:23:00,480 --> 00:23:04,888 Theater. And we have to take care of the DX in an appropriate 332 00:23:04,888 --> 00:23:11,736 way. Now remember that DX is going to be given by the XD 333 00:23:11,736 --> 00:23:14,226 theater multiplied by D theater. 334 00:23:14,370 --> 00:23:18,060 DXD theater we want to differentiate X is 10 theater 335 00:23:18,060 --> 00:23:19,536 with respect to theater. 336 00:23:20,450 --> 00:23:24,820 Now the derivative of tongue theater is the secant squared, 337 00:23:24,820 --> 00:23:27,879 so we get secret squared Theta D 338 00:23:27,879 --> 00:23:32,943 theater. So this will allow us to change the DX in here. 339 00:23:33,600 --> 00:23:40,490 Two, secant squared, Theta D Theta over on the right. 340 00:23:40,490 --> 00:23:44,550 At this stage I'm going to use the trigonometric identity, 341 00:23:44,550 --> 00:23:50,234 which says that 1 + 10 squared of an angle is equal to the 342 00:23:50,234 --> 00:23:54,700 sequence squared of the angle. So In other words, all this 343 00:23:54,700 --> 00:23:58,760 quantity down here is just the sequence squared of Theta. 344 00:23:58,780 --> 00:24:04,720 And this is very nice now because this term here will 345 00:24:04,720 --> 00:24:10,660 cancel out with this term down in the denominator down there, 346 00:24:10,660 --> 00:24:17,140 and we're left purely with the integral of one with respect to 347 00:24:17,140 --> 00:24:19,840 theater. Very simple to finish. 348 00:24:20,520 --> 00:24:24,700 The integral of one with respect to theater is just theater. 349 00:24:24,710 --> 00:24:26,390 Plus a constant of integration. 350 00:24:28,050 --> 00:24:32,910 We want to return to our original variables and if X was 351 00:24:32,910 --> 00:24:37,770 10 theater than theater is the angle whose tangent, his ex. So 352 00:24:37,770 --> 00:24:40,605 theater is 10 to the minus one 353 00:24:40,605 --> 00:24:43,749 of X. Plus a constant. 354 00:24:46,010 --> 00:24:47,770 And that's the problem finished. 355 00:24:48,290 --> 00:24:50,963 This is a very important standard result that the 356 00:24:50,963 --> 00:24:54,824 integral of one over 1 plus X squared DX is equal to the 357 00:24:54,824 --> 00:24:58,388 inverse tan 10 to the minus one of X plus a constant. 358 00:24:58,388 --> 00:25:01,358 That's a result that you'll see in all the standard 359 00:25:01,358 --> 00:25:04,031 tables of integrals, and it's a result that you'll 360 00:25:04,031 --> 00:25:07,001 need to call appan very frequently, and if you can't 361 00:25:07,001 --> 00:25:09,971 remember it, then at least you'll need to know that 362 00:25:09,971 --> 00:25:13,535 there is such a formula that exists and you want to be 363 00:25:13,535 --> 00:25:15,020 able to look it up. 364 00:25:16,720 --> 00:25:20,490 I want to generalize this a little bit to look at the case 365 00:25:20,490 --> 00:25:24,840 when we deal with not just a one here, but a more general case of 366 00:25:24,840 --> 00:25:28,320 an arbitrary constant in there. So let's look at what happens if 367 00:25:28,320 --> 00:25:30,060 we have a situation like this. 368 00:25:30,900 --> 00:25:36,900 Suppose we want to integrate one over a squared plus X squared 369 00:25:36,900 --> 00:25:38,900 with respect to X. 370 00:25:39,480 --> 00:25:42,792 Where a is a 371 00:25:42,792 --> 00:25:49,857 constant. This time I'm going to make this substitution let X be 372 00:25:49,857 --> 00:25:55,544 a town theater, and we'll see why we've made that substitution 373 00:25:55,544 --> 00:25:58,129 in just a little while. 374 00:25:58,810 --> 00:26:04,410 With this substitution, X is a Tan Theta. The differential 375 00:26:04,410 --> 00:26:08,890 DX becomes a secant squared Theta D Theta. 376 00:26:11,690 --> 00:26:14,480 Let's put all this into this 377 00:26:14,480 --> 00:26:19,705 integral here. Will have the integral of one over a squared. 378 00:26:20,980 --> 00:26:26,978 Plus And X squared will become a squared 10. 379 00:26:26,978 --> 00:26:28,451 Squared feet are. 380 00:26:29,460 --> 00:26:31,805 The 381 00:26:31,805 --> 00:26:39,434 DX Will become a sex squared Theta D 382 00:26:39,434 --> 00:26:47,256 Theta. Now what I can do now is I can take out a common 383 00:26:47,256 --> 00:26:50,208 factor of A squared from the 384 00:26:50,208 --> 00:26:57,311 denominator. Taking an A squared out from this term will leave me 385 00:26:57,311 --> 00:27:03,803 one taking a squared out from this term will leave me tan 386 00:27:03,803 --> 00:27:09,370 squared theater. And it's still on the top. I've got a sex 387 00:27:09,370 --> 00:27:10,950 squared Theta D Theta. 388 00:27:13,360 --> 00:27:20,500 We have the trig identity that 1 + 10 squared of any angle is sex 389 00:27:20,500 --> 00:27:22,404 squared of the angle. 390 00:27:22,660 --> 00:27:28,861 So I can use that identity in here to write the denominator as 391 00:27:28,861 --> 00:27:34,585 one over a squared and the 1 + 10 squared becomes simply 392 00:27:34,585 --> 00:27:36,016 sequence squared theater. 393 00:27:36,630 --> 00:27:41,442 We still gotten a secant squared theater in the numerator, and a 394 00:27:41,442 --> 00:27:45,452 lot of this is going to simplify and cancel now. 395 00:27:46,200 --> 00:27:47,652 The secant squared will go the 396 00:27:47,652 --> 00:27:52,180 top and the bottom. The one of these at the bottom will go with 397 00:27:52,180 --> 00:27:56,028 the others at the top, and we're left with the integral of one 398 00:27:56,028 --> 00:27:57,804 over A with respect to theater. 399 00:28:00,170 --> 00:28:02,890 Again, this is straightforward to finish. The integral of one 400 00:28:02,890 --> 00:28:06,426 over a one over as a constant with respect to Theta is just 401 00:28:06,426 --> 00:28:08,330 going to give me one over a. 402 00:28:08,870 --> 00:28:11,784 Theater. Plus the constant of 403 00:28:11,784 --> 00:28:16,846 integration. To return to the original variables, we've got to 404 00:28:16,846 --> 00:28:21,730 go back to our original substitution. If X is a tan 405 00:28:21,730 --> 00:28:27,058 Theta, then we can write that X over A is 10 theater. 406 00:28:27,090 --> 00:28:30,660 And In other words, that theater is the angle whose 407 00:28:30,660 --> 00:28:35,301 tangent is 10 to the minus one of all this X over a. 408 00:28:36,590 --> 00:28:41,238 That will enable me to write our final results as one over a town 409 00:28:41,238 --> 00:28:42,566 to the minus one. 410 00:28:43,250 --> 00:28:45,938 X over a. 411 00:28:46,060 --> 00:28:47,620 Plus a constant of integration. 412 00:28:49,030 --> 00:28:52,540 And this is another very important standard result that 413 00:28:52,540 --> 00:28:56,830 the integral of one over a squared plus X squared with 414 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 415 00:29:03,850 --> 00:29:07,750 constant, and as before, that's a standard result that you'll 416 00:29:07,750 --> 00:29:12,430 see frequently in all the tables of integrals, and you'll need to 417 00:29:12,430 --> 00:29:16,720 call a pawn that in lots of situations when you're required 418 00:29:16,720 --> 00:29:17,890 to do integration. 419 00:29:17,940 --> 00:29:23,940 OK, so now we've got the standard result that the 420 00:29:23,940 --> 00:29:31,140 integral of one over a squared plus X squared DX is equal 421 00:29:31,140 --> 00:29:38,340 to one over a town to the minus one of X of 422 00:29:38,340 --> 00:29:40,400 A. As a constant of integration. 423 00:29:41,040 --> 00:29:46,408 Let's see how we might use this formula in a slightly 424 00:29:46,408 --> 00:29:52,264 different case. Suppose we have the integral of 1 / 4 + 425 00:29:52,264 --> 00:29:54,216 9 X squared DX. 426 00:29:55,360 --> 00:29:58,517 Now this looks very similar to the standard formula we have 427 00:29:58,517 --> 00:30:00,770 here. Except there's a slight 428 00:30:00,770 --> 00:30:04,935 problem. And the problem is that instead of One X squared, which 429 00:30:04,935 --> 00:30:08,070 we have in the standard result, I've got nine X squared. 430 00:30:08,850 --> 00:30:11,826 What I'm going to do is I'm going to divide everything at 431 00:30:11,826 --> 00:30:15,546 the bottom by 9, take a factor of nine out so that we end up 432 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 433 00:30:19,266 --> 00:30:20,506 going to write the denominator 434 00:30:20,506 --> 00:30:25,490 like this. So I've taken a factor of nine out. You'll see 435 00:30:25,490 --> 00:30:30,050 if we multiply the brackets again here, there's 9 * 4 over 436 00:30:30,050 --> 00:30:35,370 9, which is just four and the nine times the X squared, so I 437 00:30:35,370 --> 00:30:39,550 haven't changed anything. I've just taken a factor of nine out 438 00:30:39,550 --> 00:30:45,250 the point of doing that is that now I have a single. I have a 439 00:30:45,250 --> 00:30:49,810 One X squared here, which will match the formula I have there. 440 00:30:50,450 --> 00:30:53,544 If I take the 9 outside the 441 00:30:53,544 --> 00:30:59,306 integral. I'm left with 1 /, 4 ninths plus X squared integrated 442 00:30:59,306 --> 00:31:05,546 with respect to X and I hope you can see that this is exactly one 443 00:31:05,546 --> 00:31:10,954 of the standard forms. Now when we let A squared B4 over nine 444 00:31:10,954 --> 00:31:16,778 with a squared is 4 over 9. We have the standard form. If A 445 00:31:16,778 --> 00:31:23,018 squared is 4 over 9A will be 2 over 3 and we can complete this 446 00:31:23,018 --> 00:31:27,527 integration. Using the standard result that one over 9 stays 447 00:31:27,527 --> 00:31:29,765 there, we want one over A. 448 00:31:30,540 --> 00:31:34,740 Or A is 2/3. So we want 1 / 2/3. 449 00:31:35,810 --> 00:31:37,650 10 to the minus one. 450 00:31:38,390 --> 00:31:40,238 Of X over a. 451 00:31:40,880 --> 00:31:44,800 X divided by a is X divided by 452 00:31:44,800 --> 00:31:48,238 2/3. Plus a constant of 453 00:31:48,238 --> 00:31:53,453 integration. Just to tide to these fractions up, three will 454 00:31:53,453 --> 00:31:58,040 divide into 9 three times, so we'll have 326 in the 455 00:31:58,040 --> 00:32:03,708 denominator. 10 to the minus one and dividing by 2/3 is like 456 00:32:03,708 --> 00:32:09,560 multiplying by three over 2, so I'll have 10 to the minus one of 457 00:32:09,560 --> 00:32:12,486 three X over 2 plus the constant 458 00:32:12,486 --> 00:32:16,549 of integration. So the point here is you might have to do a 459 00:32:16,549 --> 00:32:19,396 bit of work on the integrand in order to be able to write 460 00:32:19,396 --> 00:32:21,586 it in the form of one of the standard results. 461 00:32:22,870 --> 00:32:28,870 OK, let's have a look at another case where another integral to 462 00:32:28,870 --> 00:32:32,870 look at where a trigonometric substitution is appropriate. 463 00:32:32,870 --> 00:32:38,870 Suppose we want to find the integral of one over the square 464 00:32:38,870 --> 00:32:41,870 root of A squared minus X 465 00:32:41,870 --> 00:32:47,160 squared DX. Again, A is a constant. 466 00:32:49,710 --> 00:32:55,610 The substitution that I'm going to make is this one. 467 00:32:55,610 --> 00:33:00,920 I'm going to write X equals a sign theater. 468 00:33:02,130 --> 00:33:08,550 If I do that, what will happen to my integral, let's see. 469 00:33:09,080 --> 00:33:11,162 And have the integral of one 470 00:33:11,162 --> 00:33:17,976 over. The square root. The A squared will stay the same, but 471 00:33:17,976 --> 00:33:22,800 the X squared will become a squared sine squared. 472 00:33:22,800 --> 00:33:25,970 I squared sine squared Theta. 473 00:33:26,750 --> 00:33:30,794 Now the reason I've done that is because in a minute I'm 474 00:33:30,794 --> 00:33:35,175 going to take out a factor of a squared, which will leave me 475 00:33:35,175 --> 00:33:39,556 one 1 minus sign squared, and I do have an identity involving 1 476 00:33:39,556 --> 00:33:43,263 minus sign squared as we'll see, but just before we do 477 00:33:43,263 --> 00:33:47,307 that, let's substitute for the differential as well. If X is a 478 00:33:47,307 --> 00:33:51,014 sign theater, then DX will be a cosine, Theta, D, Theta. 479 00:33:52,350 --> 00:33:59,034 So we have a cosine Theta D Theta for the differential DX. 480 00:34:01,400 --> 00:34:07,388 Let me take out the factor of a squared in the denominator. 481 00:34:08,040 --> 00:34:13,837 Taking a squad from this first term will leave me one 482 00:34:13,837 --> 00:34:20,161 and a squared from the second term will leave me one minus 483 00:34:20,161 --> 00:34:21,742 sign squared Theta. 484 00:34:22,810 --> 00:34:27,094 I have still gotten a costly to the theater at the top. 485 00:34:28,490 --> 00:34:31,934 Now let me remind you there's a trig identity which says that 486 00:34:31,934 --> 00:34:35,378 the cosine squared of an angle plus the sine squared of an 487 00:34:35,378 --> 00:34:36,526 angle is always one. 488 00:34:37,160 --> 00:34:40,758 So if we have one minus the sine squared of an angle, we can 489 00:34:40,758 --> 00:34:42,043 replace it with cosine squared. 490 00:34:42,670 --> 00:34:49,633 So 1 minus sign squared Theta we can replace with simply 491 00:34:49,633 --> 00:34:51,532 cosine squared Theta. 492 00:34:51,540 --> 00:34:54,408 Is the A squared out the frontier and we want the square 493 00:34:54,408 --> 00:34:55,603 root of the whole lot. 494 00:34:56,280 --> 00:35:03,352 Now this is very simple. We want the square root of A squared 495 00:35:03,352 --> 00:35:08,792 cosine squared Theta. We square root. These squared terms will 496 00:35:08,792 --> 00:35:10,968 be just left with. 497 00:35:10,980 --> 00:35:12,600 A cosine Theta. 498 00:35:13,140 --> 00:35:18,618 In the denominator and within a cosine Theta in the numerator. 499 00:35:19,770 --> 00:35:21,960 And these were clearly cancel out. 500 00:35:23,080 --> 00:35:27,700 And we're left with the integral of one with respect to theater, 501 00:35:27,700 --> 00:35:31,165 which is just theater plus a constant of integration. 502 00:35:33,940 --> 00:35:39,232 Just to return to the original variables, given that X was a 503 00:35:39,232 --> 00:35:43,642 sign theater, then clearly X over A is sign theater. 504 00:35:44,400 --> 00:35:50,048 So theater is the angle who sign is or sign to the minus one of X 505 00:35:50,048 --> 00:35:54,990 over a, so replacing the theater with sign to the minus one of X 506 00:35:54,990 --> 00:35:57,108 over a will get this result. 507 00:35:57,650 --> 00:36:02,402 And this is a very important standard result that if you want 508 00:36:02,402 --> 00:36:07,550 to integrate 1 divided by the square root of A squared minus X 509 00:36:07,550 --> 00:36:12,302 squared, the result is the inverse sine or the sign to the 510 00:36:12,302 --> 00:36:14,678 minus one of X over a. 511 00:36:15,250 --> 00:36:16,670 Plus a constant of integration. 512 00:36:18,010 --> 00:36:25,258 Will have a look one final example which is a variant on 513 00:36:25,258 --> 00:36:31,902 the previous example. Suppose we want to integrate 1 divided by 514 00:36:31,902 --> 00:36:37,942 the square root of 4 - 9 X squared DX. 515 00:36:38,570 --> 00:36:42,902 Now that's very similar to the one we just looked at. Remember 516 00:36:42,902 --> 00:36:47,595 that we had the results that the integral of one over the square 517 00:36:47,595 --> 00:36:49,761 root of A squared minus X 518 00:36:49,761 --> 00:36:55,381 squared DX. Was the inverse sine of X over a plus a constant? 519 00:36:55,381 --> 00:36:59,291 That's keep that in mind. That's the standard result we've 520 00:36:59,291 --> 00:37:03,860 already proved. We're almost there. In this case. The problem 521 00:37:03,860 --> 00:37:08,120 is that instead of a single X squared, we've got nine X 522 00:37:08,120 --> 00:37:12,192 squared. So like we did in the other example, I'm going to take 523 00:37:12,192 --> 00:37:15,776 the factor of nine out to leave us just a single X squared in 524 00:37:15,776 --> 00:37:17,568 there, and I do that like this. 525 00:37:18,840 --> 00:37:26,060 Taking a nine out from these terms here, I'll have 526 00:37:26,060 --> 00:37:29,670 four ninths minus X squared. 527 00:37:30,290 --> 00:37:33,746 Again, the nine times the four ninths leaves the four which we 528 00:37:33,746 --> 00:37:37,202 had originally, and then we've got the nine X squared, which we 529 00:37:37,202 --> 00:37:41,910 have there. The whole point of doing that is that then I'm 530 00:37:41,910 --> 00:37:46,187 going to extract the Route 9, which is 3 and bring it right 531 00:37:46,187 --> 00:37:52,290 outside. And inside under the integral sign, I'll be left with 532 00:37:52,290 --> 00:37:58,314 one over the square root of 4 ninths minus X squared DX. 533 00:38:00,340 --> 00:38:05,464 Now in this form, I hope you can spot that we can use the 534 00:38:05,464 --> 00:38:08,758 standard result immediately with the standard results, with a 535 00:38:08,758 --> 00:38:12,052 being with a squared being equal to four ninths. 536 00:38:12,790 --> 00:38:16,696 In other words, a being equal to 537 00:38:16,696 --> 00:38:21,140 2/3. Putting all that together will have a third. That's the 538 00:38:21,140 --> 00:38:24,632 third and the integral will become the inverse sine. 539 00:38:25,940 --> 00:38:29,674 X. Divided by AA 540 00:38:29,674 --> 00:38:35,170 was 2/3. Plus a constant of integration. 541 00:38:37,140 --> 00:38:43,006 And just to tidy that up will be left with the third inverse sine 542 00:38:43,006 --> 00:38:48,453 dividing by 2/3 is the same as multiplying by three over 2, so 543 00:38:48,453 --> 00:38:52,643 will have 3X over 2 plus a constant of integration. 544 00:38:52,670 --> 00:38:56,060 And that's our final result. So we've seen a lot 545 00:38:56,060 --> 00:38:58,094 of examples that have integration using 546 00:38:58,094 --> 00:39:00,128 trigonometric identities and integration using trig 547 00:39:00,128 --> 00:39:03,179 substitutions. You need a lot of practice, and there 548 00:39:03,179 --> 00:39:06,230 are a lot of exercises in the accompanying text.