TITLE: Definite integrals with interesting results
QUESTION [15 upvotes]: I just stumbled across the fact that $\int_{-\infty}^{+\infty}{e^{-x^2}dx}=\sqrt{\pi}$. This intrigued my already-existing interest in integrals. It made me wonder, are there other integrals with crazy results?

REPLY [9 votes]: ${\tt\mbox{Sophomore's Dream}}$: Discovered by Johann Bernoulli in 1697 !!!.

$$
\int_{0}^{1}x^{-x}=\sum_{n\ =\ 1}^{\infty}n^{-n}
$$