Question

How to do annoying integrals of sines and cosines raised to powers with a weird trick.

Answer 1

So to be clear, stuff that looks like this: \[\int\limits \sin^nx \cos^mx \ dx\]

Alright, so the other day while on vacation I noticed that \[\cos^nx=\frac{1}{2^n}(e^{ix}+e^{-ix})^n\]\[\sin^nx=\frac{1}{(2i)^n}(e^{ix}-e^{-ix})^n\] and thought that it'd be nice to be able to look at this with the binomial theorem. So let's just draw out Pascal's triangle with the powers out to the side for quick and dirty reference.

So we can see that the coefficients will be from there. The only difference is the sine one will be alternating and cosine won't. For example,

\[\cos^3x=\frac{1}{2^3}(1e^{i3x}+3e^{ix}+3e^{-ix}+e^{-i3x})=\frac{1}{2^2}[1\cos(3x)+3\cos(x)]\]

Now each time you'll have this nice symmetry. Sorry IO'm on my aunt's computer and it's pretty slow sorry!