Evalutaion of a real integral using the residue theorem

[asmani]

Evaluate the following integral using the residue theorem:

http://latex.codecogs.com/gif.latex?%5Cdpi{120}%20%5Cint_{0}^{%5Cpi%20}%5Csi n^{2n}(x)%20dx

Any hint?

[lurflurf]

2i sin(x)=eix-e-ix
and
(n+1)undu=dun+1
with u=eix is
(n+1)einxdeix=dei(n+1)x

[asmani]

Sorry, I can't see how to use these facts. Can you give any further hint, please?
Besides, what contour should be chosen?

Thanks

[lurflurf]

contour is unit circle
let
z=ei x
dx=dz/(i z)
sin(x)=((z-1/z)/(2i))
sin2n(x)=((z-1/z)/(2i))2n

\int_0^\pi \sin^{2n}(x) dx=\frac{1}{2}\int_{-\pi}^\pi \sin^{2n}(x) dx=\oint_{|z|=1}\left( \frac{z-\frac{1}{z}}{2i}\right)^{2n}\frac{dz}{2i z}

[asmani]

Thanks! :smile: