Integrating $\frac{x^3}{e^x-1}$

Question

I was told that $$ \int_0^{\infty}\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}$$ in physics class.

I have no idea to compute this integral by hand. Could someone help me?

Answer 1

Expand $$\frac{x^3}{e^x-1}=\frac{x^3e^{-x}}{1-e^{-x}} =\sum_{n=1}^\infty x^3e^{-nx}$$ and then integrate termwise.