Improper integral of $\frac{x}{e^x-1}$ This integral came up in an exercise on the estimation of the specific heat of a 1-D solid and is probably a standard integral, possibly one that can be solved by contour integration:
\begin{equation}
\int_0^{+\infty} \frac{x}{e^x-1} dx
\end{equation} 
I have some rudimentary knowledge of contour integrals, but I can't come up with a proper path, also because of the many singularities along the imaginary axis. Any suggestion?

Any text book which dicusses the Riemann Zeta function will likely have this.
We have for $\Re(z) \gt 1$, that
$$ \zeta(z) \Gamma(z) = \int_{0}^{\infty} \frac{t^{z-1}}{e^t - 1} \text{d}t$$
Your integral is therefore
$$\zeta(2)\Gamma(2) = \frac{\pi^2}{6}$$
For an online discussion, see this: http://www.math.utah.edu/~milicic/zeta.pdf
