# Euler’s integral formula

This formula is well known $n! = \int_0^{+\infty} x^n e^{-x}dx$ and is given as an exercise for integration by parts. Here is a nice alternative proof I came across due to Inna Zakharevich.
$\int_0^{+\infty} e^{-\alpha x} dx = \frac{1}{\alpha}$. Now differentiate $n$ times wrt to $\alpha$ and set $\alpha = 1$.