Substitution

Why would I use a trigonometric substitution when n is even and not when n is odd?

If $n$ is even, you will get even powers of trigonometric functions, which are nice to work with because you can convert between them using the identity $\sin^{2}{x} + \cos^{2}{x} = 1$ (e.g. $\sin^{6}{x} = (1-\cos^{2}{x})^3 = ...$, so you can get an expression entirely in terms of $\cos{x}$). Odd powers of trigonometric functions are less easy to work with since you can't reduce things to only one trigonometric function in this way.

Oh thanks, I did it using constants and got to that but didn't think it was of much use as you would still get large powers and use de moivre;s theorem of something like that.