$$\int\limits_0^{ + \infty } {\frac{{{x^{2n}}}}{{{e^{\pi x}} - 1}}dx} = \frac{{\left( {2n} \right)!\zeta \left( {2n + 1} \right)}}{{{\pi ^{2n + 1}}}},{\text{ }}n \in \mathbb{N}.$$