1 00:00:00,25 --> 00:00:06,743 [MUSIC] When you're integrating powers of sines and cosines. 2 00:00:06,743 --> 00:00:13,886 I've got a motto for you to remember. You can trade sines for cosines or vice 3 00:00:13,886 --> 00:00:16,510 versa. How so? 4 00:00:16,510 --> 00:00:21,409 Well since sin squared plus cosin squared is 1, that is the Pythagorean identity, I 5 00:00:21,409 --> 00:00:27,53 can use this to get these two facts. That I can replace sin squared x by 1 6 00:00:27,53 --> 00:00:33,510 minus cosin squared, and can I replace a cosin squared by 1 minus sin squared? 7 00:00:33,510 --> 00:00:36,960 This is often useful. Let's try making some trades. 8 00:00:36,960 --> 00:00:43,450 For example, let's try to anti-differentiate say, sine cubed of x 9 00:00:43,450 --> 00:00:50,278 times cosine squared of x, dx. Our first inclination might be to try and 10 00:00:50,278 --> 00:00:55,332 make a substitution. But rats, I mean if you were sin x right 11 00:00:55,332 --> 00:01:01,687 then du would be cosine x dx. But I got a cosine squared term there 12 00:01:01,687 --> 00:01:08,162 that I have to deal with. If I made different subsitiution like, 13 00:01:08,162 --> 00:01:15,530 you know u equals cosine x or when du, would be minus sine x dx. 14 00:01:15,530 --> 00:01:19,220 But then I've got a sine cube term to deal with. 15 00:01:19,220 --> 00:01:23,772 So instead, I'll trade a pair of sines for a pair of cosines. 16 00:01:23,772 --> 00:01:26,886 So, instead of making a substitution immediately. 17 00:01:26,886 --> 00:01:30,827 I'm going to trade a pair of sines for a pair of cosines. 18 00:01:30,827 --> 00:01:35,402 What I mean, is that I'm going to rewrite the integrand as sine of x times sine 19 00:01:35,402 --> 00:01:42,363 squared of x times cosine squared of x. And then I'm going to use the fact that 20 00:01:42,363 --> 00:01:49,433 sine squared is is what? Well it's 1 minus cosine squared x. 21 00:01:51,360 --> 00:01:55,230 All right, so I can rewrite the integral as this. 22 00:01:55,230 --> 00:01:59,806 Now I can make the substitution u equals cosine x. 23 00:01:59,806 --> 00:02:08,41 So, u is cosine x and in that case du is minus sine x dx which I don't quite see 24 00:02:08,41 --> 00:02:16,271 here. But I can manufacture that by including a 25 00:02:16,271 --> 00:02:25,329 minus sign there. So this becomes the integral of 1 minus u 26 00:02:25,329 --> 00:02:34,720 squared times u squared and then minus du. 27 00:02:34,720 --> 00:02:39,710 Now I expand. So this is negative the integral of u 28 00:02:39,710 --> 00:02:43,961 squared minus u to the 4th du. Now I'll integrate. 29 00:02:43,961 --> 00:02:51,668 So this is minus and I derivative of u squared is u cubed over 3. 30 00:02:51,668 --> 00:02:59,730 And an anti derivative u to the 4th is u to the 5th over 5 plus c. 31 00:02:59,730 --> 00:03:08,402 Now I'll substitute cosine x, for u. And, we get negative cosine cubed, of x 32 00:03:08,402 --> 00:03:16,154 over 3, plus the negative of the subtraction Cosine to the 5th x over 5 33 00:03:16,154 --> 00:03:22,235 plus c. And this same kind of trick works in 34 00:03:22,235 --> 00:03:25,982 other cases too. For example, what if I wanted to 35 00:03:25,982 --> 00:03:32,420 anti-differentiate sine to the 5th power times cosine to the 5th power? 36 00:03:32,420 --> 00:03:36,680 Since I've got an odd number of cosines, I can trade all but one of them for 37 00:03:36,680 --> 00:03:41,644 sines. What I mean is I can rewrite this 38 00:03:41,644 --> 00:03:50,788 integral as sine to the 5th times cosine squared squared times cosine x dx. 39 00:03:50,788 --> 00:03:57,2 Right, this is four cosines times another cosine gives me cosine to the 5th. 40 00:03:57,2 --> 00:04:04,661 But, now, I can use the fact that cosine squared is 1 minus sine squared. 41 00:04:04,661 --> 00:04:11,829 So I can rewrite cosine squared squared, as 1 minus sine squared, squared. 42 00:04:11,829 --> 00:04:16,300 And then times cosine dx. And now we can finish by making a 43 00:04:16,300 --> 00:04:20,320 substitution. I'll make the substitution, u equals sine 44 00:04:20,320 --> 00:04:25,826 x. In that case, du is cosine x dx. 45 00:04:25,826 --> 00:04:31,820 So, the integral becomes, instead of sine to the 5th, u to the 5th. 46 00:04:31,820 --> 00:04:35,190 1 minus sine squared is 1 minus u squared. 47 00:04:35,190 --> 00:04:40,769 And that's squared. And cosine x dx is du. 48 00:04:40,769 --> 00:04:44,546 Is that going to work? Yeah, I could definitely finish this off, 49 00:04:44,546 --> 00:04:48,104 I just expand this out and I get a polynomial u. 50 00:04:48,104 --> 00:04:52,392 And I can anti-differentiate a polynomial u and then just replace u by sine of x to 51 00:04:52,392 --> 00:04:57,950 get the anti-derivative of sine of the 5th cosine of the 5th. 52 00:04:57,950 --> 00:05:01,210 So what's the general pattern to this kind of trick? 53 00:05:01,210 --> 00:05:04,360 The trick works as long as we've got an odd power on the sine, or an odd power on 54 00:05:04,360 --> 00:05:08,290 the cosine. 'Cuz in that case, I can split off all 55 00:05:08,290 --> 00:05:12,748 but one of them. And then I'll get say, sine times an even 56 00:05:12,748 --> 00:05:16,810 power of sine, times some number of cosines. 57 00:05:16,810 --> 00:05:21,690 Or sine, times an even number of cosines, times cosine. 58 00:05:21,690 --> 00:05:30,99 And since this an even number here, I can rewrite those in terms of the other. 59 00:05:30,99 --> 00:05:36,247 So then I'll end up with a single sine times a bunch of cosines, really a 60 00:05:36,247 --> 00:05:43,821 polynomial and cosine x. Or a polynomial and sine x times cosine, 61 00:05:43,821 --> 00:05:50,304 and then I can finish it off with a single substitution.