Evalutaion of a real integral using the residue theorem

[asmani]

Evaluate the following integral using the residue theorem:

Any hint?

 

[lurflurf]

2i sin(x)=e^ix-e^-ix
and
(n+1)uⁿdu=duⁿ⁺¹
with u=e^ix is
(n+1)e^inxde^ix=de^i(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=e^{i 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!