Maths C4 - Integration... Help???

My teacher did teach us the reverse chain rule however he stressed very hard that you can only use it when function inside brackets is linear. Don't know if that's fully true

My teacher did teach us the reverse chain rule however he stressed very hard that you can only use it when function inside brackets is linear. Don't know if that's fully true but good so that it that rule doesn't even pass through my mind during c4 and I just use it when doing c3.

Assuming you mean (f)

Look at your formula book and see which of these terms you can integrate. Then let us know which terms you're still stuck on.

Thank you! For some reason I forgot I could use the identity that $1 + cot^2 \theta \equiv cosec^2 \theta$

So I managed to get...

$= \int (cot^2 x - 2 cot x cosec x + cosec^2 x ) dx$

$= \int (cosec^2 x - 1 - 2 cot x cosec x + cosec^2 x ) dx$

$= -2cot x - x + 2cosec x + c$

Ok, so now I'm stuck on how to integrate these types...

$\int (3cos x sin^2 x) dx$

Ok, so now I'm stuck on how to integrate these types...

$\int (3cos x sin^2 x) dx$

I tell right away it will be $\sin^3(x)+c$ but whenever you see these types, just realise that cosine and sine are a product here, and they are both derivatives of each other (ignoring the sign change which is a constant mult.). So it would be a good idea to use substitution. Which one do you think would be a good one to sub in for?

Any thoughts?

If you have no idea then try differentiating $\sin^3 x$ then try $\sin^4 x$ and $\sin^5 x$. You should notice a pattern (you may know this pattern already).

Getting used to these patterns will save you having to make a substitution in the exam (which may or may not be given to you). And I find that if you discover the pattern yourself as I've told you to above then it helps you remember it.

How on earth did you know instantly? I'm really struggling with these types I'm not even sure I know how to use Integration by Substitution in this case



Ok, so I can see (worked out) that...
$\int sin^3 x = 3 cosx sin^2 x$

$\int sin^4 x = 4 cosx sin^3 x$

$\int sin^5 x = 5 cosx sin^4 x$ << I didn't check whether this one was correct I just assumed.

How on earth did you know instantly? I'm really struggling with these types I'm not even sure I know how to use Integration by Substitution in this case



Ok, so I can see (worked out) that...
$\int sin^3 x = 3 cosx sin^2 x$

$\int sin^4 x = 4 cosx sin^3 x$

$\int sin^5 x = 5 cosx sin^4 x$ << I didn't check whether this one was correct I just assumed.

Use the one which has the higher exponent on it, so sine has the higher exponent. Thus $u=\sin(x)$ works.

Also, those integral signs should be derivative signs

Yes so this shows you if you need to integrate something of the form $\cos x$ multiplied by a power of $\sin x$ then you can "reverse the chain rule" by raising the power of the $\sin x$ by 1, ignoring coefficients for now. Can you see this?

Similarly you could integrate $\sin x \cos^6 x$ by considering the derivative of $\cos^7 x$.

It takes a lot of practice to get confident with this type of integration. Please keep asking if you need help.

Yes I'm not sure why I used $\int$ instead of $f'(x)$ haha, can you tell my brain is fried?

Awesome, I'll try using the substitution that $u = sin(x)$ now

Thank you!



Yes I'm starting to see it now. I never properly came across these types until now so I guess I need to carry on familiarising myself with them!

Thank you so much!

Yes I'm not sure why I used $\int$ instead of $f'(x)$ haha, can you tell my brain is fried?

Awesome, I'll try using the substitution that $u = sin(x)$ now

Thank you!

Yes I'm starting to see it now. I never properly came across these types until now so I guess I need to carry on familiarising myself with them!

Thank you so much!

...

Why $n\geq 1$?

By the way, this integral pattern isn't mentioned in the spec but is in the Edexcel textbook. And I've seen a few past paper questions that can be done almost instantly if you are confident with this integral form.

What is mentioned on the spec are integrals like

$\displaystyle \int \frac{3x}{2x^2+4} \ dx$

Again you can use a substitution but the best method here is to recognise the pattern.

For fun you can try proving that $\displaystyle \int (n+1)\cos(x)\sin^n(x) .dx = \sin^{n+1}(x)+c$ where $n \geq 1$ using substitution. I found it always nice to test my knowledge at A-Level by considering the general examples.

Then switch 'em around and prove that $\displaystyle \int (n+1)\sin(x)\cos^n(x).dx = -\cos^{n+1}(x)+c$ where $n \geq 1$ using an appropriate substitution

Might help you become more used to the pattern here and these problems in general.

Extension:

I'm not even sure I even know how to solve $\displaystyle \int \frac{3x}{2x^2+4} \ dx$ using Integration by Substitution let alone by recognition

I'm not even sure I even know how to solve $\displaystyle \int \frac{3x}{2x^2+4} \ dx$ using Integration by Substitution let alone by recognition

Feel like giving up.

I think I know how to integrate by using the fact that... $\int \frac{f'(x)}{f(x)} dx = ln |f(x)| + c$



Yes I definitely have to give this a go!

You really need to know how to do these.

You need to spot that the numerator is a constant away from the derivative of $2x^2+4$. And when you spot this pattern, the integral is always $\ln (\text{the denominator})+c$ and then you adjust the constant in front of the log.

You should know this since the derivative of $\ln f(x)$ is $\frac{f'(x)}{f(x)}$ so if you are integrating a fraction where the numerator is the derivative of the denominator then by reversing the differentiation you end up with $\ln f(x)$.