Definite integrals with interesting results [closed]

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?

One I just evaluated

$$\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2} = \frac{2}{3}$$

or the ultimate insanity...

$$\int_{-1}^1\frac{dx}{x}\sqrt{\frac{1+x}{1-x}}\log\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) = 4 \pi \operatorname{arccot}{\sqrt{\phi}} $$

where $\phi=(1+\sqrt{5})/2$ is the golden ratio.

This one, which is another version of your integral$$ \int_{-\infty}^{+\infty}e^{-\left(x+ \tan x \right)^2} \mathrm{d}x=\sqrt{\pi},$$ or this strange family $$ \begin{align} \displaystyle \int_0^{1} \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi (\left\{1/x\right\}+1)}}}} \mathrm{d}x & = \dfrac{F_{n-1}}{F_{n}} - \dfrac{(-1)^{n}}{F_{n}^2} \ln \left( \dfrac{F_{n+2}-F_{n}\gamma}{F_{n+1}-F_{n}\gamma} \right),\\\\ \end{align} $$ where $\left\{x\right\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$, $\gamma$ is the Euler constant, $F_{n}$ are the Fibonacci numbers, $\psi:=\Gamma'/\Gamma$ is the digamma function and where the continued fraction has $n$ horizontal bars.

${\tt\mbox{Sophomore's Dream}}$: Discovered by Johann Bernoulli in 1697 !!!.

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