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Why would I use a trigonometric substitution when n is even and not when n is odd?
If 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 (e.g. , so you can get an expression entirely in terms of ). 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.