An integral related to the zeta function at the point $2$ is given by
$\zeta(2) = \int\nolimits_0^\infty \dfrac{t}{e^t - 1}\mathrm dt$
How to calculate this integral?
An integral related to the zeta function at the point $2$ is given by
$\zeta(2) = \int\nolimits_0^\infty \dfrac{t}{e^t - 1}\mathrm dt$
How to calculate this integral?
The integrand can be expressed as a geometric series with first term $te^{-t}$ and common ratio $e^{-t}$. Integrate term-by-term (after justifying it, of course) and see if you don't recognize the result as $\zeta(2)$.
Somewhat equivalent to Gerry's answer: let $t=-\log(1-u)$, giving the integral
$-\int_0^1 \frac{\log(1-u)}{u}\mathrm du$
Expand the logarithm as a series, swap summation and integration, and then you should be able to see something familiar...