# How to evaluate this integral to get pi^2/6:

by hb1547
Tags: evaluate, integral, pi2 or 6
 P: 35 $\int_0^\infty \frac{u}{e^u - 1}$ I know that this integral is $\frac{\pi^2}{6}$, just from having seen it before, but I'm not really sure if I can evaluate it directly to show that. I know that: $\zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du$ Does the value $\frac{\pi^2}{6}$ come from using other methods of showing the result for $\zeta(2)$ and solving the equation, or is that integral another way of evaluating $\zeta(2)$?
 Quote by hb1547 $\int_0^\infty \frac{u}{e^u - 1}$ I know that this integral is $\frac{\pi^2}{6}$, just from having seen it before, but I'm not really sure if I can evaluate it directly to show that. I know that: $\zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du$ Does the value $\frac{\pi^2}{6}$ come from using other methods of showing the result for $\zeta(2)$ and solving the equation, or is that integral another way of evaluating $\zeta(2)$?