A integral consists of sin(n+1/2) n=1,2,3

[xhole]

Homework Statement

Hi all, i encounter this integral while i am trying to find the Fourier series of f(x)=ln(2*sin(x/2))


pi sin[(n+1/2)x] sin[(n-1/2)x]
∫ ------------- ＋ ------------- dx n=1,2,3...
0 sin[(1/2)x] sin[(1/2)x]

Homework Equations

sin(A+B)=sinAcosB+sinBcosA
sin(2A)=2sinAcosA

The Attempt at a Solution

the answer seems to be 2*pi, but i still don't know how to work it out..

 

[snipez90]

It looks like a job for the Dirichlet Kernel:

http://en.wikipedia.org/wiki/Dirichlet_kernel