PF Integral Bee: Share Interesting/Quirky Integrals!

[ergospherical]

Thought it could be fun to have a sort of "PF Integral Bee"... if you know some interesting/quirky/etc. integrals then post them here!

To get the ball rolling...
1. $\displaystyle{\int_0^1} \dfrac{\ln{(x+1)}}{x^2+1} dx$

 

[mcastillo356]

Not studied integrals seriously, but for me it has been really fascinating to understand $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$. The demonstration is similar to the demonstration of fundamental theorem of calculus.