Integral Examples 1

00:01 So, let's go straight into our examples and apply what we've just learned.

00:05 Let's look at our first example.

00:09 We are looking at the integral, a 5 sin of x dx.

00:17 Now, remember that this time, you are integrating.

00:20 Let me just remind you about this. When we did sin of x and we differentiated it, we went to cos of x.

00:28 When we had cos of x and we differentiated it, we got minus sin of x.

00:33 So, let's just say this direction is differentiation.

00:38 If you want to integrate, all you do is go backwards.

00:43 So, if I go in that direction and that direction, I will be integrating it.

00:48 So, in some sense, a little table like this might help for you to remember because in some questions, you'll find that we'll be differentiating and integrating trig at the same time.

01:00 So, it gets pretty complicated and quite easy to make mistakes.

01:04 So, sin differentiates to cos. Cos differentiates to minus sin.

01:09 So, cos integrates to sin and sin would, this sin here would obviously integrate to minus cos because of the minus there.

01:18 Okay, so, let's come back to our example.

01:21 So, fairly straightforward when you're integrating, five is just a constant.

01:26 There's no x next to it. So, that can just stay as it is.

01:29 Sin x, so, look at sin x here and we're going backwards because you want to integrate, so, we're going in this direction.

01:36 So, if minus sin x goes to cos x, if you didn't have the minus there, sin x would go to minus cos x.

01:43 So, this goes to minus cos of x and remember, you have a plus c at the end.

01:49 You get rid of the integral sin just to show that you have integrated, so, that's just for us to remember and you have now found the general integral of 5 sin of x dx to be minus 5 cos of x plus c.

02:03 Let's look at our next question. We are now building up from the question that we just did.

02:09 We have 5 sin (10x) dx. Let me just write what I wrote earlier.

02:17 If I just put s instead of sin and that for cos and then, I have this.

02:24 Try and land this little table because I find it really helps. The s stands for sin. That's cos.

02:29 C stands for cos and that stands for sin. If I go in this direction, I'm differentiating.

02:35 If I go in the opposite direction, I'm integrating.

02:40 So, putting that on the side is really going to help me speed up and I don't have to think too much about what sin and cos differentiate or integrate to.

02:49 So, let's try this out here. Five is a constant, so, that can stay.

02:55 Observe this closely. This is a function and you've got an inside function here.

03:01 So, you have the chain rule, well, not the chain rule, chain rule when we differentiate, we're doing the opposite of the chain rule. So, remember what we do.

03:09 When you use the chain rule, you differentiated the outside, you multiplied it with the differential of the inside.

03:16 This time, we integrate the outside and we divide by the differential.

03:21 So, sin here, because we're going in the integration direction, integrates to minus cos.

03:27 So, this can go to minus cos of 10x and don't forget, you have a function on the inside.

03:34 This time, instead of timesing it with the differential, we divide it with the differential plus c.

03:41 This cancels a little bit, so, I've got one and two, so, by five times table, leaving me with cos of 10 x over 2 plus c.

03:54 So, we've done some examples with sin's and cos's and also, when you have a function inside of a function and we're integrating it.

04:02 Here, we now have E's. So, let's have a play around with how you integrate an E function.

04:08 The question is asking you to do e to the 6x dx. In order to do this, I have to remind you firstly of how to differentiate E.

04:18 So, remember, we said that e of x differentiate to e of x, so, it differentiates in that direction but it also integrates to the same thing.

04:27 So, that makes it nice and easy.

04:28 The other thing to remember is that if I was differentiating e to the 6x, the differential would be the chain rule, so, you've got a function which is just e of 6x, so, that stays as it is. And then, you times it with the differential of the inside function.

04:48 When you integrate, you integrate the entire outside function first and then, you divide it by the differential of the inside.

04:57 So, let's see if we can apply that here. Integrate E to the power of anything.

05:02 It will stay the same.

05:04 So, if it's e to the power of 6x, when you integrate it, it stays the same as e to the power of 6x.

05:11 So, I've integrated the outside function.

05:15 I now need to divide it by the differential of the inside function which in our case is 6x.

05:21 The differential of six is six, and then, you add c to it.

05:25 So, when you integrate e to the 6x dx, your answer is e to the 6x, divided by 6 and then, plus c.