nt.number theory - Integral expression for zeta(2)

The starting point is the integral

$$
Gamma(s) = int_{0}^{infty}e^{-x}x^{s-1}dx
$$

for the gamma function. Make the change of variable $x = nu$ with $n$ an arbitrary positive integer. Then

$$
Gamma(s)n^{-s} = int_{0}^{infty}e^{-nu}u^{s-1}du
$$

and summing over $n$ from $n = 1$ yields

$$
Gamma(s)zeta(s) = int_0^{infty}frac{1}{e^u - 1}u^{s-1}du.
$$

This formula was the starting point of one of Riemann's two proofs of the functional equation. I am not certain who discovered it first, but it may have been Abel.

Substituting $s = 2$ gives

$$
zeta(2) = int_{0}^{infty}frac{u}{e^u - 1}du
$$

and so

$$
zeta(2) =
int_{0}^{infty}frac{ue^u}{e^{2u} - e^u}du =
int_{0}^{infty}frac{u(e^u - 1) + u}{e^{2u} - e^u}du =
int_{0}^{infty}left(ue^{-u} + frac{u}{e^{2u} - e^u}right)du =
1 + int_{0}^{infty}frac{u}{e^{2u} - e^u}du.
$$