1 00:00:00,000 --> 00:00:05,054 [music] How do I antidifferentiate trig functions? 2 00:00:05,055 --> 00:00:10,700 Sine and cosine are easy if we just remember their derivatives. 3 00:00:10,700 --> 00:00:15,800 The derivative of sine of x, with respect to x, is cosine x. 4 00:00:15,800 --> 00:00:19,980 And the derivative of negative cosine is sine. 5 00:00:19,980 --> 00:00:24,782 So what are their anti-derivatives? Knowing this, now I know some of the 6 00:00:24,782 --> 00:00:30,570 anti-derivatives. If the derivative of sine is cosine, then 7 00:00:30,570 --> 00:00:35,533 the anti-derivative of cosine is sine plus C. 8 00:00:35,533 --> 00:00:39,766 And if the derivative of negative cosine is sine. 9 00:00:39,766 --> 00:00:45,044 That means the anti-derivative of sine is negative cosine plus c. 10 00:00:45,044 --> 00:00:50,472 Well what about tangent? What would I differentiate to get tangent 11 00:00:50,472 --> 00:00:53,796 of x? Yeah, I'm asking what do I differentiate 12 00:00:53,796 --> 00:00:57,953 to get tangent or what's the anti-derivative of tangent? 13 00:00:57,954 --> 00:01:05,014 I'm claiming, that the antiderivative of tangent is log secant x, plus some 14 00:01:05,014 --> 00:01:08,164 constant. How do I know this? 15 00:01:08,164 --> 00:01:13,060 Well, to verify this is true, it's enough to do a differentiation problem. 16 00:01:13,060 --> 00:01:19,343 I need to check that the derivative of log of secant of x Is equal to tan of x. 17 00:01:19,344 --> 00:01:23,000 And how do I know this? Well I have to differentiate this 18 00:01:23,000 --> 00:01:26,310 composition of functions. The composition of log and secant,. 19 00:01:26,310 --> 00:01:31,046 Which I'll do in the chain rule, right. So what's the derivative of log? 20 00:01:31,046 --> 00:01:34,656 It's one over. So it's 1 over the inside function times 21 00:01:34,656 --> 00:01:38,519 the derivative of secant. What's the derivative of secant? 22 00:01:38,520 --> 00:01:44,212 It's secant tangent. Alright, so this is 1 over secant times 23 00:01:44,212 --> 00:01:49,309 secant x times tangent x. Now good news the secants cancel. 24 00:01:49,310 --> 00:01:55,512 And what I'm left with is just tangent x. So I've found an antiderivative for 25 00:01:55,512 --> 00:01:59,435 tangent. But it's totally opaque as to how somebody 26 00:01:59,435 --> 00:02:03,544 would ever have gotten that as the anti-derivative of tangent. 27 00:02:03,545 --> 00:02:07,506 And yet, once that candidate anti-derivative was revealed to us, we can 28 00:02:07,506 --> 00:02:12,126 verify that it's truly an anti-derivative just by differentiating it and checking 29 00:02:12,126 --> 00:02:15,751 that it really gives us tangent. Ok, let's try another one of these kinds 30 00:02:15,751 --> 00:02:18,229 of problems. What's an anti-derivative of secant? 31 00:02:18,230 --> 00:02:21,814 So I'm asking for a function, whose derivative is secant x. 32 00:02:21,815 --> 00:02:24,124 Right? I'm looking for the antiderivative of 33 00:02:24,124 --> 00:02:25,887 secant x. What is that? 34 00:02:25,888 --> 00:02:29,761 Well it turns out that the antiderivative of secant x is this. 35 00:02:29,761 --> 00:02:33,330 It's log, the absolute value of secant x plus tangent x. 36 00:02:33,330 --> 00:02:36,770 Now again, it's totally mysterious, right, where this came from. 37 00:02:36,770 --> 00:02:39,906 But we can verify that that's truly an antiderivative. 38 00:02:39,906 --> 00:02:44,191 So to verify that the antiderivative of secant x is this, it's enough to 39 00:02:44,191 --> 00:02:46,817 differentiate this and end up with secant x. 40 00:02:46,817 --> 00:02:52,405 So let's try that, alright? I'll differentiate log of the absolute 41 00:02:52,405 --> 00:02:56,303 value of secant x plus tangent x and what do I get? 42 00:02:56,304 --> 00:03:01,914 Well, the derivative of log is 1 over the inside function, which is secant x plus 43 00:03:01,914 --> 00:03:06,122 tangent x. Times the derivative of the inside 44 00:03:06,122 --> 00:03:08,924 function. This is really the chain rule in action. 45 00:03:08,924 --> 00:03:12,839 All right. Now, what's the derivative of this sum? 46 00:03:12,839 --> 00:03:16,406 Well, that's the sum of derivatives, right? 47 00:03:16,406 --> 00:03:20,453 So the derivative of secant is secant x tangent x. 48 00:03:20,453 --> 00:03:30,041 And the derivative of, tangent. Is secant squared, and this is divided by 49 00:03:30,041 --> 00:03:37,612 secant x plus tangent x. Now note I can factor out a secant x here, 50 00:03:37,612 --> 00:03:44,064 and I've got secant x times the numerator is tangent x plus secant x. 51 00:03:44,065 --> 00:03:48,253 The denominator's the same thing. I mean I wrote it in the opposite order. 52 00:03:48,253 --> 00:03:53,430 But now this is great, right? This cancels and what I'm left with is 53 00:03:53,430 --> 00:03:57,478 secant x. So yeah, the derivative of this is secant 54 00:03:57,478 --> 00:04:02,010 x and consequently the antiderivative of secant x is this. 55 00:04:02,010 --> 00:04:05,731 This is really a key in to anti multiplying, to factoring. 56 00:04:05,731 --> 00:04:10,051 If you remember our experience, multiplying and anti-multiplying, or 57 00:04:10,051 --> 00:04:12,914 factoring numbers, right. Factoring was hard. 58 00:04:12,914 --> 00:04:16,754 I had to do a bunch of trial division to figure out how to factor a number. 59 00:04:16,754 --> 00:04:20,979 But once I had that factorization, or purported factorization, in hand, then 60 00:04:20,979 --> 00:04:24,944 checking it was easy. Multiplying is something that you can do. 61 00:04:24,945 --> 00:04:28,024 It's the same story here with derivatives, right. 62 00:04:28,025 --> 00:04:31,675 Anti differentiating is hard, I mean it seems hard to figure out what the anti 63 00:04:31,675 --> 00:04:35,771 differentiating derivative is. But once you show me a guess, I can verify 64 00:04:35,771 --> 00:04:40,571 that your guess is correct by doing the easy step of just differentiating and 65 00:04:40,571 --> 00:04:44,434 seeing if you're right. I hope that these examples are giving you 66 00:04:44,434 --> 00:04:47,554 some real respect for the true difficulties of anti-differentiation. 67 00:04:47,554 --> 00:04:52,606 Differentiating is a mechanical process. You just have to apply the rules 68 00:04:52,606 --> 00:04:54,354 carefully. But we've seen now that 69 00:04:54,354 --> 00:04:58,842 anti-differentiation requires flashes of insight, just cooking up the correct 70 00:04:58,842 --> 00:05:01,452 answer and then verifying that it's correct. 71 00:05:01,452 --> 00:05:04,492 Well this seems, you know, impossible to continue, right? 72 00:05:04,492 --> 00:05:08,852 How can we ever find anti-derivatives if we just have to keep hoping for insight to 73 00:05:08,852 --> 00:05:11,510 appear? Well the good news is that most of the 74 00:05:11,510 --> 00:05:16,190 rest of this course will be developing tools to make this kind of guessing more 75 00:05:16,190 --> 00:05:19,790 systematic right? Instead of relying on just sheer insight 76 00:05:19,790 --> 00:05:23,213 to solve these problems. We're going to be providing tools that 77 00:05:23,213 --> 00:05:26,753 will aid your insight or aid your creativity, or provide a little bit of 78 00:05:26,753 --> 00:05:30,593 structure so that you can exploit the patterns that we're going to be seeing, 79 00:05:30,593 --> 00:05:33,320 and order the finite derivatives much more easily. 80 00:05:33,320 --> 00:05:44,233 So I think it seems hard right now, but it's going to get a lot easier.