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.

    Author of lecture Integral Examples 1

     Batool Akmal

    Batool Akmal

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