Integration: Exercise 1

00:01 So our first example here gives you two functions that are multiplying together and we're having to integrate them.

00:08 You have X that is multiplying with cos of 5X.

00:13 Now, the first thing for you is to just look at this question and to observe everything that's happening.

00:18 We are integrating.

00:19 You have an X function.

00:21 You have a cos of 5x function.

00:24 Cos of 5x is a function of a function, so you have a little function inside of it as well.

00:30 So make these decisions before you start.

00:33 You also might think, "Maybe I should do this using substitution." You could take a substitution of U equals to 5X.

00:41 But because we said we are going to do this with integration by parts just to practice, let’s continue using that formula.

00:48 So integration by part states that you have uv dashed equals to uv minus the integral of the vdu dx.

01:00 Now we need everything.

01:01 We need a U and a U dashed, a V dashed and a V.

01:07 Remember, these dashed terms are just dU by dX if you want it to write that in full and this is just dV by dX.

01:14 But just because there’s a quite few complicated notations going on here, we just stick with the dashed notation because it makes it easier.

01:22 So let’s split this into U and V.

01:24 If we go for this as U and this as V dashed, hopefully this should work.

01:32 If you think that it’s starting to make this function bigger, we may have to swap them.

01:38 So let’s differentiate U to just give me 1.

01:42 So remember that this function is going to be differentiated and this function is going to be integrated.

01:48 We said that V dashed is cos of 5X.

01:52 So we now need to calculate what V is.

01:55 A quick reminder of differentiating and integrating trigonometry, so sine and cos and cos goes to minus sine, so when you differentiate, sine is going to cos, so go in that direction.

02:07 Sine goes to cos and cos goes to minus sine and when you integrate, go in the other direction.

02:13 So let me just call that differentiation and integration.

02:16 When you integrate, cos goes to sine and sine goes to minus cos.

02:21 You may just be able to learn this, but sometimes when you're doing this together, it helps to just have something visual on the side that can remind you what direction you’re going in.

02:31 So remember, I’m integrating cos.

02:33 I want to go here and cos integrates to positive sine.

02:37 So this just goes to sine 5X.

02:41 We now need to do something with the differential of the inside.

02:44 Because you’re integrating, you don’t multiply it.

02:47 You divide it with the differential of the inside and that’s your V.

02:53 Now that we have everything, let’s put this all into the formula.

02:56 I want uv, so those 2 functions multiplied together.

03:00 That gives me x multiplied by sine 5x.

03:04 Put that in bracket, so it’s clearer -- over 5 and then I minus the integral of vdu, dx.

03:10 So it’s those 2 terms here.

03:13 My U dashed is just 1 and my V is sine of 5X over 5, and you see that we have obtained our objective which was to make this integral easier.

03:26 So this is now in terms of 1 function of X rather than 2 functions of X.

03:31 We just tidy this up.

03:33 So I can write this as X over 5, I just move this right here.

03:36 It doesn’t matter where it is, multiplied by sine of 5X.

03:41 I then minus the integral of, let’s take this 5 out, 1 over 5, sine of 5x, dx.

03:51 And remember, you’re looking at integrating sine.

03:53 So this is your sine, and you integrate, you go backwards.

03:56 This is going to go minus cos because there is a minus here.

04:01 It’s getting a little bit messy there, but there is a minus there.

04:03 So sine will integrate to minus cos.

04:06 So lastly, I have x over 5, sine of 5x minus 1 over 5, stays as it is.

04:14 Sine now integrates to minus cos of 5x and don’t forget, this is a function inside of a function.

04:23 So we also have to divide it with 5.

04:27 I’ll put my plus C right at the end once I finish.

04:30 So that becomes x over 5, sine of 5x.

04:35 The minus and the minus makes this function plus.

04:38 The 5 and the 5 gives you a 25 at the bottom and then you have cos of 5x at the top and we plus C.