The Series $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{r^2} + \cdots$

In an appendix to a Physics book, I found the derivation on infinite integrals of the type $$\int_0^\infty \frac{x^n}{e^x-1}dx.$$