TITLE: Integrating $ \int_0^\infty \frac{x^5}{e^x+1} \, dx $
QUESTION [6 upvotes]: This improper integral has stumped me and not many integration problems give me problems. However, this one made me think over my limit. Finally, as I could not even get started on this problem, I thought to post it here, and have some talented mathematicians look at it. The problem is to integrate:
$$  \int_0^\infty \frac{x^5}{e^x+1} \, dx $$

REPLY [2 votes]: \begin{align}
\int_0^\infty \frac{x^5}{e^x+1}\, dx&=\int_0^\infty x^5 \Big(\frac{1}{e^x-1}-\frac{2}{e^{2 x}-1}\Big)\, dx\\
&=\int_0^\infty  \frac{x^5}{e^x-1} \, dx-2\int_0^\infty  \frac{x^5}{e^{2 x}-1} \, dx\\
&=\int_0^\infty  \frac{x^5}{e^x-1} \, dx-2^{-5}\int_0^\infty  \frac{x^5}{e^{ x}-1} \, dx\\
&=(1-2^{-5})\int_0^\infty  \frac{x^5}{e^{ x}-1} \, dx\\
&=(1-2^{-5})\Gamma(6)\zeta(6)\\
\end{align}
note that $\int_0^\infty  \frac{x^{s-1}}{e^{ x}-1}\,dx=\Gamma(s)\zeta(s)$.
In general then it holds that $$\int_0^\infty \frac{x^s}{e^x+1} \, dx=(1-2^{-s})\Gamma(s+1)\zeta(s+1)$$