Maths C4 - Integration... Help???

Your teacher is correct and taught the topic well. So many teachers don't which is why we get so many students on TSR who think that e.g.

$\displaystyle \int e^{x^2} \ dx = \frac{1}{2x} e^{x^2}+c$

Ok so all of the integrals can be worked out without the need of having to use Substitution? So Integration by Parts is still in there etc?

I ask because I'm thinking of using it on...

$\int 2x(x^2-3)^4 dx$

Substitute $u=\sin(3x-2)$

Or recognise that it's almost in the form $f'(x)[f(x)]^n$

Oh yes I'm still having trouble using $f'(x)[f(x)]^n$ with these types e.g. $f(x) = sin^3 x \Rightarrow f'(x) = 3 cosx sin^2 x$

So I managed to work out...
$\int cos(3x-2)sin^2(3x-2) dx$

$= \frac{1}{3} \int 3cos(3x-2)sin^2(3x-2) dx$

$= \frac{1}{9}sin^3 (3x-2) + c$


Gonna attempt this by substitution now I will not be defeated!

Edit: @RDKGames Ok, maybe I actually am still confused with the example above! How is $\int cos x sin^2 x$ classified as a $f'(x)[f(x)]^n$ type if the derivative of $sin^2 x$ is $2cos x sinx$ ??

You pick $\sin(x)$ not $\sin^2(x)$

Wait... I think I see it now, it's cos $\int cos x sin^2 x$ can be re-written as $\int cos x (sinx)^2$ so that fits the $f'(x)[f(x)]^n$ criteria, right?

Yes.

You're not too confident with the pattern recognition / inspection methods of integration so you're going to find a lot of the integrals on that sheet very challenging. But actually these recognition/inspection methods aren't required that much in C4 exams. So don't make this the focus of your revision - keep doing as many past papers as you can.

If you can spot the pattern then that's the main thing since once you've spotted it you can either reverse the differentiation (the quickest method) or you can make a substitution for the function whose derivative you can also see in the expression.

E.g. $\displaystyle \int 3x(4x^2+3)^5 \ dx$

You need to notice that $3x$ is a constant away from the derivative of $4x^2+3$ so that means you make a substitution for $4x^2+3$. This works because in the substitution method you find the derivative of $4x^2+3$ and this ends up "cancelling" with the $3x$.

Or the quickest method is to consider the derivative of $(4x^2+3)^6$ but this takes a lot of practice.

Yeah I'm definitely still trying to get to grips with $f'(x)[f(x)]^n = \frac{[f(x)]^{n+1}}{n+1} + c$

It may take me a while

This is what inspection is, in essence.

So I just attempted Question 5 of Edexcel C4 January 2013 paper and came across a few silly little things I have forgotten such as the fact that $\frac{d(a^x)}{dx} = a^x ln a$

But then I came across something that I can't recall ever seeing in my life! $\int a^x dx = \frac{a^x}{ln a}$
Why have I never come across this before?!!

They don't come up often, and I think they're in the formula booklet, so it doesn't matter whether you know them or not. Have a go at deriving it yourself so you how it's true and be able to derive it yourself in the exam.

They're not in the formula book for Edexcel (unless I'm being stupid). I know how to prove $\frac{d(a^x)}{dx} = a^x ln a$ but I have no idea how to prove $\int a^x dx = \frac{a^x}{ln a}$

Reality is starting to hit me that I'm not prepared for these exams over the next 2 months

So I just attempted Question 5 of Edexcel C4 January 2013 paper and came across a few silly little things I have forgotten such as the fact that $\frac{d(a^x)}{dx} = a^x ln a$

But then I came across something that I can't recall ever seeing in my life! $\int a^x dx = \frac{a^x}{ln a}$
Why have I never come across this before?!!

Yes when this integral came up it stumped a lot of students.

If you know that the derivative of $a^x$ is $a^x \ln a$ then you should be able to reverse this to find the integral of $a^x$ since $\ln a$ is just a constant.

Alternatively, you are given this result in the formula book:

$a^x = e^{x\ln a}$

You can use this to find both the derivative and integral of $a^x$. Let us know if you're not sure how.

But for the derivative of $a^x$ it's worth also knowing the proof that involves taking logs of both sides / implicit differentiation since there's a small chance that it would be required in the exam.

SHOULD be able to but I don't think I understand how I can prove $\frac{d(a^x)}{dx} = a^x ln a$ but I don't know how to prove $\int a^x dx = \frac{a^x}{ln a}$ FML why is everything so difficult for me to get my head around? :/

Also I have no idea how $a^x = e^{x\ln a}$ is linked to the above Seriously feel like giving up this close to the exams!

Well... you know how to integrate $e^{ax}$ where $a$ is a constant, so you already know how to integrate $e^{x\ln(a)}$

So if...
$\int e^{2x} = \frac{1}{2} e^{2x}$

then...
$\int e^{xln(a)} = \frac{1}{ln(a)} e^{xln(a)}$ ??? Something tells me I did something wrong there!

Why do you think its wrong?

Cos it's not the same format as $\int a^x dx = \frac{a^x}{ln a}$

I actually have no idea what I'm talking about right now