[Introverted moron]

I don't get integrating using standard patterns at all. Can someone please explain this to me (preferably using harder examples from the relevant exercise in the C4 book)?

[Plato's Trousers]

(Original post by Introverted moron)
I don't get integrating using standard patterns at all. Can someone please explain this to me (preferably using harder examples from the relevant exercise in the C4 book)?

Basically, there are certain types of function that have standard integrals, so you look at your function and see if it fits any of those patterns and you can find the integral quite easily without having to work it out from scratch. For example:



(where a is a constant)

So let's say you had



your a is 3, so the answer would be



This is just a simple example, but there's a good list of standard integrals here. The trick is recognising whether your function is in the form of one of the ones on the list and then writing in that form. Then it's easy!

[Introverted moron]

(Original post by Plato's Trousers)
Basically, there are certain types of function that have standard integrals, so you look at your function and see if it fits any of those patterns and you can find the integral quite easily without having to work it out from scratch. For example:



(where a is a constant)

So let's say you had



your a is 3, so the answer would be



This is just a simple example, but there's a good list of standard integrals here. The trick is recognising whether your function is in the form of one of the ones on the list and then writing in that form. Then it's easy!

I can't do easy.

How do you integrate: sec(^2)x tan (^2)x?

Sorry, I would use Latex, but am feeling a bit lazy at the moment.

I recognise that sec x differentiates to sec x tan x and I've tried doing it in the way they've done it in the book i.e. letting y = sec (^2) x but that only differentiates to 2 sec(^2) tan x.

.......or maybe I'm just doing it all wrong. :/

[Goldfishy]

(Original post by Introverted moron)
I can't do easy.

How do you integrate: sec(^2)x tan (^2)x?

Sorry, I would use Latex, but am feeling a bit lazy at the moment.

I recognise that sec x differentiates to sec x tan x and I've tried doing it in the way they've done it in the book i.e. letting y = sec (^2) x but that only differentiates to 2 sec(^2) tan x.

.......or maybe I'm just doing it all wrong. :/

Try the sub u = tanx

[Introverted moron]

(Original post by Goldfishy)
Try the sub u = tanx

Show me please?

[Goldfishy]

(Original post by Introverted moron)
Show me please?

Have you covered integration by substitution yet?

If so, then in this specific question it starts like this:

Let

So becomes Which is simple to integrate. Then sub back x into the result.

A note on the choice of sub

The reason for choosing u = tanx is that when differentiating u wrt x you get sec^2 x which also appears in the integrand. Since you have a tan^2 x term in the integrand as well, then it is a recognisable case of chain rule. (Personally, I prefer doing it by inspection). I think this is a case of experience in choosing the right sub to do, but usually they give you the sub they want you to do

[Introverted moron]

(Original post by Goldfishy)
Have you covered integration by substitution yet?

If so, then in this specific question it starts like this:

Let

So becomes Which is simple to integrate. Then sub back x into the result.

A note on the choice of sub

The reason for choosing u = tanx is that when differentiating u wrt x you get sec^2 x which also appears in the integrand. Since you have a tan^2 x term in the integrand as well, then it is a recognisable case of chain rule. (Personally, I prefer doing it by inspection). I think this is a case of experience in choosing the right sub to do, but usually they give you the sub they want you to do

Thank you for your help.

We have done integration by substitution, but it's the topic after the integration using standard patterns one, so I assume that it's possible to do it using standard patterns.

Managed to work it out using standard patterns this morning using y = tan(^3)x, differentiating that and then adjusting.

......but now I'm stuck on another one.



I know tan(^2)x differentiates to sec(^2)x, which could be useful.......or might not be?

[Goldfishy]

(Original post by Introverted moron)
Thank you for your help.

We have done integration by substitution, but it's the topic after the integration using standard patterns one, so I assume that it's possible to do it using standard patterns.

Managed to work it out using standard patterns this morning using y = tan(^3)x, differentiating that and then adjusting.

......but now I'm stuck on another one.



I know tan(^2)x differentiates to sec(^2)x, which could be useful.......or might not be?

Yep, that way is fine too, and is in effect how you know what kind of sub to use.

For the new one, expand the brackets and you should get two recognisable integrals.

[Akbar2k7]

I did not get this too the examples in the book are so bad thanks guys.

[ztibor]

(Original post by Goldfishy)
Yep, that way is fine too, and is in effect how you know what kind of sub to use.

For the new one, expand the brackets and you should get two recognisable integrals.

For differenciating use the chain rule.
First of all, this function is a power function but the variable is a function (tan x)
According to the rule, after differentiating as power function, we have to multiply
by the derivative of inner function(tan x->sec^2x)
So

For integration:

Generally