1 00:00:05,90 --> 00:00:09,954 We already know how to handle some integration problems where the integrand 2 00:00:09,954 --> 00:00:15,918 is a powers of sines and cosines. For example, I can anti-differentiate 3 00:00:15,918 --> 00:00:19,520 sine to and odd power. How? 4 00:00:19,520 --> 00:00:23,800 By trading in all but one of those sins for cosines. 5 00:00:23,800 --> 00:00:26,572 Let's make this really concrete. Instead of talking about an odd power 6 00:00:26,572 --> 00:00:30,558 well let's just make it 17. Can I antidifferentiate sin to the 17th 7 00:00:30,558 --> 00:00:34,42 power? Yeah, I can rewrite this problem as sin 8 00:00:34,42 --> 00:00:38,659 of x times sin squared x to the 8th power, because sin square to the 8th 9 00:00:38,659 --> 00:00:43,600 power gives me 16 copies of sin times more sin gives me 17 copies of sin, now 10 00:00:43,600 --> 00:00:48,622 we can trade sin squared for some cos signs right, sin squared is 1 minus cos 11 00:00:48,622 --> 00:00:58,489 sign squared to the 8th power. So if I can do this anti-differentiation 12 00:00:58,489 --> 00:01:04,380 problem all I need to do is do this anti-differentiation problem. 13 00:01:04,380 --> 00:01:11,426 And I can do that with a substitution. So make the substitution u equals cosine 14 00:01:11,426 --> 00:01:17,320 x and in that case d u is minus sign x dx. 15 00:01:17,320 --> 00:01:20,344 Now maybe you're complaining you don't see a minus sign. 16 00:01:20,344 --> 00:01:23,524 And you'll only see sin, but I'll just put a pair of cancelling minus sins 17 00:01:23,524 --> 00:01:26,651 there, and I see a minus sin next dx so we got a du and the rest I can write in 18 00:01:26,651 --> 00:01:31,776 terms of u. Specifically, this becomes negative, the 19 00:01:31,776 --> 00:01:36,54 n is derivative of 1 minus u squared to the 8 power du, is what's left over and 20 00:01:36,54 --> 00:01:40,608 this anti-differentiation problem that I can do and it is a little bit annoying 21 00:01:40,608 --> 00:01:45,93 cause I guess I gotta expand this thing out to find an anti-derivative of this, 22 00:01:45,93 --> 00:01:53,75 but I can do it. But what if instead I'd had an even power 23 00:01:53,75 --> 00:01:57,482 of sine. Well, maybe I'll anti-differentiate sine 24 00:01:57,482 --> 00:01:59,900 to the fouth power. All right? 25 00:01:59,900 --> 00:02:02,930 So this is not an odd power but an even power. 26 00:02:02,930 --> 00:02:07,214 that's harder, I can't trade in all of my sins for cosines because I need just one 27 00:02:07,214 --> 00:02:13,190 left over to make the substitution work. Instead, I'll use an identity, the half 28 00:02:13,190 --> 00:02:17,165 angle formula. It let's me replace sined squared of x by 29 00:02:17,165 --> 00:02:22,978 1 minus cosine of 2x over 2. And it lets me replace cosign squared of 30 00:02:22,978 --> 00:02:28,510 x by 1 plus cosign 2x over 2. How do those help? 31 00:02:28,510 --> 00:02:35,481 Well I can rewrite this integral as the integral of sin squared, squared. 32 00:02:35,481 --> 00:02:38,260 And now I can use this half angle identity. 33 00:02:38,260 --> 00:02:41,770 This is the same as the integral of what sin squared. 34 00:02:41,770 --> 00:02:48,100 Sin squared is 1 minus cos 2x over 2. All right? 35 00:02:48,100 --> 00:02:51,686 These are equal. And still have to square that, dx. 36 00:02:51,686 --> 00:02:55,686 Now, I expand. So, I get this is the same as 37 00:02:55,686 --> 00:03:01,572 integrating. Well, the cos 2x over 2 term, squared is 38 00:03:01,572 --> 00:03:08,370 cosigned squared of 2x over 4. And there's our cross term. 39 00:03:08,370 --> 00:03:13,640 The 1 half times minus cosign 2 x over 2. And there's 2 of those. 40 00:03:13,640 --> 00:03:20,60 So I end up getting minus 1 half. cosigned 2x. 41 00:03:20,60 --> 00:03:24,911 And there's the one half term squared just plus a quarter, so I just have to do 42 00:03:24,911 --> 00:03:30,990 this integration problem. I can split that into 3 integrals. 43 00:03:30,990 --> 00:03:38,293 So this gives me the integral of cos sign squared 2x over 4dx minus the integral of 44 00:03:38,293 --> 00:03:45,760 cos sign 2x over 2dx plus the integral of 1 4th dx. 45 00:03:45,760 --> 00:03:50,286 Now, the second and third integration problem I can do, I'm just going to copy 46 00:03:50,286 --> 00:03:56,502 down this one again. The integral of cos squared 2x over 4 dx. 47 00:03:56,502 --> 00:04:00,500 This, here, well I could do this by making a substitution. 48 00:04:00,500 --> 00:04:05,484 u equals 2x, say. And then I'll get that this is sine of 2x 49 00:04:05,484 --> 00:04:12,972 divided by 4. And if I integrate a quarter, I get 1 4th 50 00:04:12,972 --> 00:04:16,180 x. What about that first one? 51 00:04:16,180 --> 00:04:20,946 How do I integrate cosine squared? So, to handle this, I can repeat the 52 00:04:20,946 --> 00:04:25,945 trick with the half angle identity. But I'm looking at integrating cosign 53 00:04:25,945 --> 00:04:30,73 squared 2x. So I'll replace x by 2x and I'll turn 54 00:04:30,73 --> 00:04:34,520 this into a 4x. So I can use this identity. 55 00:04:34,520 --> 00:04:38,770 How does that go? Well I get instead of this the integral 56 00:04:38,770 --> 00:04:44,350 of 1 plus cosine 4x and instead of over 4 its now over 8 and then I'll just copy 57 00:04:44,350 --> 00:04:54,310 down these things again. Sin 2x over 4 plus a quarter x. 58 00:04:54,310 --> 00:04:59,240 Now I can put it all together. So I want to and differentiate 1 8th. 59 00:04:59,240 --> 00:05:04,126 I get 1 8th x. And then I want to, and I differentiate 60 00:05:04,126 --> 00:05:12,754 cos 4x over 8, and I get sin of 4x over 32 and then I'll include the rest. 61 00:05:12,754 --> 00:05:20,125 So, I'll subtract sin of 2x over 4. I'll add a quarter x and I'll add some 62 00:05:20,125 --> 00:05:26,222 constant. And I could combine the one quarter x and 63 00:05:26,222 --> 00:05:33,662 the 1 8x, so I could right this as 3 8x plus sine of 4x over 32 minus sine of 2x 64 00:05:33,662 --> 00:05:41,130 over 4 plus c. I should say that in some cases you can 65 00:05:41,130 --> 00:05:46,639 get away with doing a bit less work. So I want to calculate the integral from 66 00:05:46,639 --> 00:05:52,117 zero to pi of sine to the fourth and I'll again start the same way, right I'm going 67 00:05:52,117 --> 00:06:01,150 to integrate from zero to pi and I'll write this as sine squared squared. 68 00:06:01,150 --> 00:06:04,134 Just so I can see how the half angle formula is going to help me. 69 00:06:04,134 --> 00:06:10,24 Alright, now I'll use the half angle formula as the integral from zero to pi 70 00:06:10,24 --> 00:06:17,658 of 1 minus cosine2x over 2 squared, but now what do I want to do? 71 00:06:17,658 --> 00:06:22,923 Well, I'll expand that out again, so this is the integral from 0 to pi of 1 fourth 72 00:06:22,923 --> 00:06:28,269 Minus the cross term is one half cosign 2x and then plus the cosign times squared 73 00:06:28,269 --> 00:06:33,800 2x over 4. And now there's a little trick. 74 00:06:33,800 --> 00:06:41,794 I'm integrating from 0 to pi cosign. Right I'm integrating cosign over an 75 00:06:41,794 --> 00:06:44,770 entire period. And that ends up being 0. 76 00:06:44,770 --> 00:06:49,630 So I can just throw this whole term away and now I can keep on going. 77 00:06:49,630 --> 00:06:52,760 I can also use the half angle formula here. 78 00:06:52,760 --> 00:06:58,427 So this is the integral from 0 to pi. well I've still got the 1/4 plus, and 79 00:06:58,427 --> 00:07:02,917 then what does this become? By the half angle formula, this is 1 80 00:07:02,917 --> 00:07:05,288 plus. And instead of 2x. 81 00:07:05,288 --> 00:07:10,553 It's cosine 4x over 8, but I would again, if I integrate cosine 4x, x going from 0 82 00:07:10,553 --> 00:07:18,598 to pi, that's integrating cosine ove 2 complete periods that ends up cancelling. 83 00:07:18,598 --> 00:07:23,458 So I can just throw that term away, and all I'm really integrating now is a 84 00:07:23,458 --> 00:07:29,309 quarter plus an 8. Well thats 3 8ths, but I'm integrating 85 00:07:29,309 --> 00:07:36,410 over an interval of life pi, so this definite integral is 3 8ths pi. 86 00:07:36,410 --> 00:07:40,41 This turn out to be not so bad. See I'm getting an answer of 3 8ths pi, 87 00:07:40,41 --> 00:07:45,125 but that's not really the point right? The cool thing about setting this up as a 88 00:07:45,125 --> 00:07:51,395 definite integral is really just how easy it is to do the calculation since I can 89 00:07:51,395 --> 00:08:00,773 throw away some terms along the way that I know would integrate to zero.