Table of Integrals and Derivatives

00:01 So we’ve just looked at basic integration.

00:04 We’ve learned how integration is different from differentiation.

00:08 We’ve also learned that integration helps us find the area and their curves.

00:12 And we’ve learned the mathematical method of adding one to the power and dividing by the new power.

00:18 Like we did in differentiation where then we moved on to finding out how to differentiate trig functions.

00:24 And so, we’re going to do exactly the same with integration.

00:26 We’re going to start to look at trigonometric functions, and we’re going to look at some different methods of integration when you deal with more complicated functions.

00:38 Let me just talk you through what we’re about to do.

00:41 We’re going to look at trigonometric integration.

00:44 So I’ll teach you the standard results of lots of different trig functions.

00:49 We’re also going to do integration by substitution, which is a different type of method of integration, makes life a little bit easier.

00:56 And then the last one is going to be integration by parts, which is fairly complicated functions, and we’re going to talk about this new technique, which is splitting it into parts to integrate.

01:07 The techniques are obviously going to use a lot of trig functions and the new techniques that we’re about to learn.

01:13 And the application, you know about this already.

01:15 We use integration to either solve differential equations or to find the area and their curves.

01:22 So let’s just look at some standard results that we’ve previously looked at.

01:26 We are now talking about differentiation.

01:28 So, when we differentiated, if you had a function f of x, and that was sine(x), you remember that that differentiated to cos(x).

01:37 We also derived this result using differentiation from first principles.

01:44 If you’re looking at a function cos(x), that differentiates to -sine(x), and we’ve done this so many times.

01:50 Hopefully, you’ve just managed to learn this.

01:53 Tan(x) differentiates to sec squared of x.

01:56 And again, we looked at the proofs for this in the differentiation lectures.

02:01 Now, the next few, you don’t have to learn.

02:03 They’re just easy to derive if you need to.

02:06 But just to give you the standard results, if you’re ever looking for them in your formula booklets or textbooks, cot(x) differentiates to -cosec squared of x.

02:15 Sec(x) goes to sec(x)tan(x).

02:19 Cosec of x differentiates to -cosec(x)cot(x).

02:25 And these last two, we needed to learn.

02:27 So remember, we evaluated that e of x differentiates to e of x, so it doesn’t change.

02:33 And ln of x differentiates to 1 over x.

02:37 Now, if you think about what I said in the previous lecture, integration is just differentiation backwards.

02:43 So you could look at all the derivative f' of x column here and go backwards.

02:50 And if then you integrated cos of x, so just look at the first line, if you integrated cos of x, you would get sine of x.

02:57 If you integrated -sine of x, you would get cos of x.

03:01 So instead of going in this direction, you’re going in the backward direction.

03:06 Let me just show that to you in a table so it makes more sense.

03:11 So here’s what I’m trying to say.

03:13 I’m reversing the table that we just looked at.

03:17 If you are asked to integrate cos of x, that goes to sine of x.

03:22 If you’re asked to integrate sine of x, this now goes to -cos of x.

03:27 Do you remember what cos of x differentiated to? -sine of x, and that’s where that minus comes from.

03:34 It gets a little bit tricky so we’re going to try and just write something down for ourselves so we can remember.

03:40 So once again, cos of x integrates to sine of x.

03:43 And sine of x, this time, integrates to -cos of x.

03:48 Sec squared of x goes to tan of x, just the way tan of x differentiated to sec squared of x.

03:54 Sec squared of x integrates to tan of x.

03:58 Nice and easy, e of x never changes.

04:00 So of e of x differentiated to e of x; e of x integrates to e of x.

04:06 And remember what we said when you differentiate ln of x, you get 1 over x.

04:11 So when we integrate 1 over x, we’re going to go back to ln of x.

04:16 There’s quite a few results here to familiarize yourselves with, but we’ll keep this at the front or on the side somewhere so that we can see it as we integrate.

04:24 Remember, this will all come with practice.

04:27 So we’re going to do a lot of examples now just to practice these new integrals.