Integration

How do I integrate 3^(2x)? Or any constant to the power of x?

Do you know the derivative of a^x?

Yes, a^xlna

How do I integrate 3^(2x)? Or any constant to the power of x?

Before anyone starts suggesting substitution, just write $3^{2x} = 9^x$

Now I leave you in the good hands of Notnek.

Okay so that tells you that the integral of $a^x \ln a$ is $a^x + c$. Can you use this to find the integral of $a^x$?

It's also worth memorising this integral because it comes up in exams. But I recommend trying to derive it first, plus this will be useful if you forget it in an exam!

I'd assume it's a^x /lna + c, since if you differentiate it the lna and 1/lna would cancel, right?

I think it's fine to use reverse chain rule here if the student is confident with it - it should be easy if the student understands what they're doing

Yes that's right. You should have that on your list of standard integrals that you should learn off by heart. The intgeral of a^x often catches students out in exams.

So it would be 3^(2x) /2ln3?

I've only just seen it today in a past exam question, my teachers briefly skimmed over C4 differentiation/integration so I've probably missed a lot of things I ought to know! Glad I found it before the exam.

Make sure you know how to integrate $\sin^2 x$ and $\cos^2 x$ because they come up all the time. Also something similar to $\int \sin^2 x \cos x$ seems to come up frequently.

Yeah, I just change them using the cos double angle formula, and I'm familiar with the inspection type questions, it'd just be 1/3sin^3 x + c. Thank you very much