integration - Integrating the Dirichlet Kernel using complex analysis

I was looking at the Dirichlet integral
$\frac{1}{2\pi}\int_{-\pi}^{\pi}{(sin((N+\frac{1}{2})x))\over sin(\frac{x}{2})}dx$
and it seems to be possible to integrate using complex analysis. I attempted this using the residue theorem but my understanding of complex analysis is still quite shallow and I was unable to derive an answer. Is there any way to do this integral with complex analysis?