Definite integrals with interesting results 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?
 A: $\displaystyle \int_0^{+\infty}\dfrac{\sin x}{x}dx=\dfrac{\pi}{2}$
$\displaystyle \int_0^{+\infty}\cos(x^2)dx=\dfrac{1}{2}\sqrt{\dfrac{\pi}{2}}$
Math.stackexchange is full of such beautiful formulae.
A: 
${\tt\mbox{Sophomore's Dream}}$: Discovered by Johann Bernoulli in 1697 !!!.

$$
\int_{0}^{1}x^{-x}=\sum_{n\ =\ 1}^{\infty}n^{-n}
$$
A: $$\large\int_{-\infty}^\infty\, \frac{1}{1+(x+\tan x)^2}\, dx= \pi
$$
A: $$\begin{align}
\int_0^\infty\exp\Big(-\sqrt[n]x\Big)dx&=n!
\\
\\
\int_0^1\Big(1-\sqrt[n]x\Big)^mdx&={m+n\choose n}^{-1}={m+n\choose m}^{-1}
\\
\\
\int_0^1\ln\Big(1-\sqrt[n]x\Big)dx&=-H_n
\\
\\
\lim_{n\to\infty}n\Big(1-\sqrt[n]x\Big)&=-\ln x
\end{align}$$
A: $$\int_{\mathbf{R}}\frac{dx}{1+x^2}=\pi$$
and 
$$\frac{22}{7} - \pi = \int_0^1  \frac{(x-x^2)^4}{1+x^2}dx.$$
A: $$\int_0^{\infty}\left(\frac{3}{4}\right)^{\lfloor{x}\rfloor}\ dx=4$$
and
$$\int_{-\infty}^{\infty}e^{-|x|}\ dx=2$$
A: Well I think $$\displaystyle \begin{align*} \int_{0}^{\frac{\pi}{2}} \theta \log^{3} (\tan\theta) \, d\theta &= \frac{7 \pi^{2}}{32} \zeta (3) + \frac{93}{16} \zeta (5). \end{align*}$$
is pretty interesting.
A: 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.
A: 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.
