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## Homework Statement

Use contour integrals and justify your steps, find [tex]\int[/tex] cos

^{2n}x dx where the integral is from 0 to 2pi.

## Homework Equations

## The Attempt at a Solution

My first thought was that this integral would be zero, using the residue theorem with residue 0, as I'm integrating round a closed curve. I think this is wrong but I don't know why.

I then decided to write f(x) = cos

^{2n}x = [e

^{ix}+ e

^{-ix}/ 2i]

^{2n}

I could use the substitution z=e

^{ix}to get f(z) = [1/2i (z + 1/z)]

^{2n}, this would have a singularity at z=0 but as z is an exponential it is never 0.

I thought about using a keyhole contour with f(z) as above, but I'm really not sure.