1 00:00:00,005 --> 00:00:07,911 [music] We've already seen anti-derivatives for sine and for cosine, 2 00:00:07,911 --> 00:00:13,408 so what's an anti-derivative for say, cosine 3x? 3 00:00:13,409 --> 00:00:17,064 Right? I'm looking for some function that 4 00:00:17,064 --> 00:00:21,486 differentiates to cosine 3x. Well, there's basically one trick that 5 00:00:21,486 --> 00:00:24,833 we've been using throughout all of these anti-differentiation problems. 6 00:00:24,833 --> 00:00:29,154 It's guessing. Sine differentiates to cosine, so maybe 7 00:00:29,154 --> 00:00:33,401 sine of 3x is a good guess. But if I differentiate sine of 3x using 8 00:00:33,401 --> 00:00:38,852 the chain rule, I get the derivative of the outside function, that's looking good, 9 00:00:38,852 --> 00:00:42,428 at the inside function, that's looking good, too. 10 00:00:42,428 --> 00:00:46,703 But times the derivative of the inside function which is 3. 11 00:00:46,703 --> 00:00:50,653 So that's no good, but I can fix that pretty easily, right? 12 00:00:50,653 --> 00:00:57,804 If I just include a 1 3rd here, that rigs up everything perfectly, right? 13 00:00:57,804 --> 00:01:05,485 Because then the 1 3rd here and the 3 cancel, and I am left with cosine of 3x. 14 00:01:05,485 --> 00:01:14,924 So, I can summarize this by saying that an anti-derivative for cosine 3x is 1 3rd 15 00:01:14,924 --> 00:01:17,330 sine 3x. And then, to get in the other 16 00:01:17,330 --> 00:01:20,529 anti-derivative, I just add some locally constant function c. 17 00:01:20,530 --> 00:01:25,458 Well, that worked out. Let's try to do a slightly harder example. 18 00:01:25,458 --> 00:01:30,386 Let's try to anti-differentiate say, sine of 2x plus 1. 19 00:01:30,386 --> 00:01:37,397 Alright, I'm looking for some function which differentiates to 2x plus 1. 20 00:01:37,397 --> 00:01:40,185 I might guess that the answer will involve cosine. 21 00:01:40,186 --> 00:01:44,020 Well, I know that cosine differentiates to minus sine. 22 00:01:44,020 --> 00:01:48,211 So, my first guess might be cosine of 2x plus 1. 23 00:01:48,211 --> 00:01:55,075 Because if I differentiate this, I get minus sine of the inside function, times 24 00:01:55,075 --> 00:02:00,794 the derivative of the inside function. So, this is not exactly this, so I haven't 25 00:02:00,794 --> 00:02:05,665 quite found an anti-derivative. But, it'll be easy to patch that up, 26 00:02:05,665 --> 00:02:09,211 right? If I instead look at a slightly different 27 00:02:09,211 --> 00:02:13,115 function. If I include a factor of minus 1 half in 28 00:02:13,115 --> 00:02:19,983 front of cosine of 2x plus 1, then the derivative would be minus 1 half times the 29 00:02:19,983 --> 00:02:25,760 derivative of cosine 2x plus 1, which is minus sine 2x plus 1 times 2. 30 00:02:25,760 --> 00:02:32,130 And now, the good news is that the minus 1 half, and this minus sign and this 2 31 00:02:32,130 --> 00:02:38,206 cancel, and what I'm left with then is just sine of 2x plus 1 which means I've 32 00:02:38,206 --> 00:02:45,347 found an anti-derivative, right? An anti-derivative for sine of 2x plus 1 33 00:02:45,347 --> 00:02:54,447 is, according to this argument, negative 1 half cosine 2x plus 1, plus some constant 34 00:02:54,447 --> 00:02:57,010 c. I can see a bit of a general principle 35 00:02:57,010 --> 00:02:58,790 here. Well, here's our set up. 36 00:02:58,790 --> 00:03:02,660 I've got m and b, just a couple of constants. 37 00:03:02,660 --> 00:03:08,541 And what I want to do is anti-differentiate a function evaluated at 38 00:03:08,541 --> 00:03:11,786 mx plus b. Alright, this is the general framework 39 00:03:11,786 --> 00:03:16,684 that's general enough to encompass the examples we've done thus far, right? 40 00:03:16,684 --> 00:03:22,302 F could be say, sine or cosine. In our first example, we looked at cosine 41 00:03:22,302 --> 00:03:27,522 of 3x, so m was 3 and b was 0. In our second example, we looked at sine 42 00:03:27,522 --> 00:03:30,310 of 2x plus 1, so m was 2 and b was 1 and f was sine. 43 00:03:30,310 --> 00:03:34,531 But, this is sort of a general example that encompasses some of the specific 44 00:03:34,531 --> 00:03:37,206 example which we're already taking a look at. 45 00:03:37,206 --> 00:03:40,867 Let's suppose that you already have an anti-derivative for f. 46 00:03:40,868 --> 00:03:47,501 Yeah, let's suppose that I know how to anti-differentiate just f, and let's call 47 00:03:47,501 --> 00:03:52,355 that F, right? So, the derivative of F is f. 48 00:03:52,356 --> 00:03:55,416 Now, how do I deal with the mx plus b part? 49 00:03:55,417 --> 00:03:58,875 Well basically, we're trying to guess. What I'm trying to find some function that 50 00:03:58,875 --> 00:04:02,479 differentiates to this, and I've got a function that just differentiates to just 51 00:04:02,479 --> 00:04:04,920 f. So, let's see what happens if I 52 00:04:04,920 --> 00:04:08,850 differentiate F of mx plus b by using the chain rule. 53 00:04:08,850 --> 00:04:16,374 Well, the derivative of F is f, evaluated the inside times the derivative of the 54 00:04:16,374 --> 00:04:22,000 inside, which is just m. Now, I can fix this if I include a factor 55 00:04:22,000 --> 00:04:25,836 of 1 over m here. Then, I get a factor of 1 over m here. 56 00:04:25,836 --> 00:04:30,158 And that's great because then this 1 over m, and this m cancel. 57 00:04:30,158 --> 00:04:34,477 So here, I've got a function whose derivative is f of mx plus b. 58 00:04:34,478 --> 00:04:41,574 In other words, the anti-derivative of f of mx plus b is this, right? 59 00:04:41,575 --> 00:04:48,574 It's F of mx plus b divided by m plus c. Instead of always just relying on guessing 60 00:04:48,574 --> 00:04:52,291 and hoping that we can guess the correct anti-derivative, we're going to be 61 00:04:52,291 --> 00:04:55,380 learning more and more general techniques along these lines. 62 00:04:55,380 --> 00:04:59,090 The big one will go by the name substitution, or the chain rule in 63 00:04:59,090 --> 00:05:03,340 reverse. And as we develop more and more of these 64 00:05:03,340 --> 00:05:09,780 anti-differentiation techniques in our tool box, we'll be able to 65 00:05:09,780 --> 00:05:17,485 anti-differentiate a whole lot more functions without relying on just flashes 66 00:05:17,485 --> 00:05:18,700 of insight.