Evalutaion of a real integral using the residue theorem

  • Thread starter asmani
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  • #1
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Main Question or Discussion Point

Evaluate the following integral using the residue theorem:

gif.latex?%5Cdpi{120}%20%5Cint_{0}^{%5Cpi%20}%5Csin^{2n}(x)%20dx.gif


Any hint?
 

Answers and Replies

  • #2
lurflurf
Homework Helper
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2i sin(x)=eix-e-ix
and
(n+1)undu=dun+1
with u=eix is
(n+1)einxdeix=dei(n+1)x
 
  • #3
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Sorry, I can't see how to use these facts. Can you give any further hint, please?
Besides, what contour should be chosen?

Thanks
 
Last edited:
  • #4
lurflurf
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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

[tex]\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}[/tex]
 
  • #5
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Thanks! :smile:
 

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