This may be aimed at PH(or whomever), since he is up on CA and I am just learning it a little as I find time.

How could we go about using complex integration to solve this integral from a previous post?. Assuming it's worthwhile in this case. Are cases where there are infinite poles more difficult?.

$\displaystyle \int_{0}^{\infty}\frac{x}{e^{x}-1}dx$

I see it has poles at $\displaystyle 2c{\pi}i$ and a double at 0, I think.

Should we end up with something like $\displaystyle 2{\pi}i(\frac{\pi}{12i})=\frac{{\pi}^{2}}{6}$

Or maybe even $\displaystyle 2{\pi}i(-\frac{\pi}{12}i)=\frac{{\pi}^{2}}{6}$