# Integral Examples 1

by Batool Akmal

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00:02 So, let’s go straight into our examples and apply what we’ve just learned.

00:07 Let’s look at our first example.

00:10 We are looking at the integral of 5 sin of x, dx.

00:18 Now, remember that this time, you are integrating.

00:21 Let me just remind you about this.

00:23 When we did sine 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 -sine of x.

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

00:39 If you want to integrate, all you do is go backwards.

00:42 So if I go in that direction and that direction, I will be integrating it.

00:50 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 sine differentiates to cos, cos differentiates to -sine.

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

01:19 Okay.

01:19 So let’s come back to our example.

01:22 So, fairly straightforward when you’re integrating, 5 is just the constant.

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

01:29 Sine(x), so look at sine(x) here, and we’re going backwards because you want to integrate, so we’re going in this direction.

01:36 So if -sine(x) goes to cos(x), if you didn’t have the minus there, sine(x) would go to -cos(x).

01:43 So this goes to -cos of x.

01:47 And remember, you have a +c at the end.

01:49 You get rid of the integral sine just to show that you have integrated.

01:53 So, that’s just for us to remember.

01:55 And you have now found the general integral of 5sine of x dx to be -5cos of x + c.

02:04 Let’s look at our next question.

02:06 We are now building up from the question that we just did.

02:09 We have 5sine(10x)dx.

02:16 Let me just write what I wrote earlier.

02:18 If I just put s instead of sine, and that for cos, and then I have this.

02:24 Try and lend this little table because I find it really helps.

02:27 The s stands for sine, that’s cos.

02:30 C stands for cos, and that stands for sine.

02:32 If I go in this direction, I’m differentiating.

02:36 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 sine and cos is, differentiate or integrate to.

02:50 So, let’s try this out here.

02:52 Five is a constant, so that can stay.

02:55 Observe this closely.

02:57 This is a function and you’ve got an inside function here.

03:01 So, you have the chain rule.

03:03 Well, not the chain rule; chain rule when we differentiate.

03:06 We’re doing the opposite of the chain rule.

03:08 So remember what we do.

03:10 When you used 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, sine here, because we’re going in the integration direction, integrates to -cos.

03:28 So this can go to -cos of 10x.

03:32 And don’t forget, you have a function on the inside.

03:35 This time, instead of timesing it with the differential, we divide it with the differential, + c.

03:42 This cancels a little bit, so I’ve got 1 and 2.

03:45 So by 5 times table, leaving you with cos of 10x over 2 + c.

03:55 Things are getting a little bit more exciting now, so let’s have a look at this.

03:58 We can see that this is a function inside of a function, and we can also see some trig.

04:04 So, let’s try this out.

04:07 I have the integral of 1 + cosx cubed dx.

04:15 I’m just going to put that little table or sine or coses on the side.

04:18 So remember what we said, sine goes to cos; cos minus sine; this direction is differentiation, this direction is integration.

04:27 Hopefully, these letters are starting to make more sense.

04:30 Okay.

04:31 So we’re starting straight away with a big function and a little function.

04:36 We are saying that we are integrating it, so be very careful.

04:39 With so much calculus, it can often get quite confusing with differentiation and integration happening so close to each other.

04:48 So, when you integrate a function like this, you’d integrate the outside function first.

04:53 So, you can rewrite this as 1 + cosx, don’t change the inside, to the power of 4.

04:58 Add 1 to the power, divide by new power.

05:01 And now, you also divide by the differential of the inside.

05:06 So you may think, because we’re doing an integration question, everything isn’t being integrated.

05:11 But we now need to divide it with the differential of the inside.

05:15 So 1 + cosx, when you differentiate 1, it just disappears.

05:19 Let’s look at what cos differentiates to.

05:21 So differentiation is in that direction here.

05:23 So there’s my cos.

05:25 So cos differentiates to -sine.

05:27 So we’re going to have -sine(x) there.

05:30 I haven’t put my + c yet.

05:32 I’ll do that right at the end.

05:33 So that gives me to the power of 4 over -4sine(x), or you can make this entire answer negative and write it as 1 + cosx to the power of 4 over 4sine(x) + c.

05:50 So we’ve done some examples with sines and coses, and also when you have a function inside of a function and we’re integrating it.

05:57 Here, we now have e's.

05:59 So let’s have a play around with how you integrate an e function.

06:04 The question is asking you to do e to the 6xdx.

06:08 In order to do this, I have to remind you firstly of how to differentiate e.

06:13 Remember, we said that e of x differentiates to e of x.

06:17 So it differentiates in that direction, but it also integrates to the same thing.

06:22 So, that makes it nice and easy.

06:24 The other thing to remember is that if I was differentiating e to the 6x, the differential would be the chain rule.

06:34 So you got a function which is just e of 6x, so that stays as it is.

06:39 And then you times it with the differential of the inside function.

06:44 When you integrate, you integrate the entire outside function first, and then you divide it by the differential of the inside.

06:52 So let’s see if we can apply that here.

06:55 Integrate e to the power of anything.

06:58 It will stay the same.

07:00 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.

07:08 So I’ve integrated the outside function.

07:10 I now need to divide it by the differential of the inside function, which in our case is 6x.

07:16 The differential of 6 is 6, and then you add c to it.

07:21 So, when you integrate e to the 6xdx, your answer is e to the 6x divided by 6 and then plus c.

### About the Lecture

The lecture Integral Examples 1 by Batool Akmal is from the course Advanced Integration.

### Included Quiz Questions

1. 7cos(x)+c
2. -7cos(x)+c
3. 7cos(x)
4. -7cos(x)
5. -7sin(x)+c
1. [4cos(x) / a]+c
2. 4cos(x)+c
3. 4acos(x)+c
4. [-4cos(x) / a]+c
5. -4acos(x)+c
1. x+[cos(7x) / 7]+c
2. x+[cos(x) / 7]+c
3. x+cos(7x)+c
4. x - [cos(7x) / 7]+c
5. x+[cos(x) / 7]
1. [e^(bx) / b] + c
2. [e^(bx) / b]
3. e^(bx) + c
4. be^(bx) + c
5. -be^(bx) + c

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