Euler’s integral formula

June 10, 2015 · by randomprojection · in Calculus · 1 Comment

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 $\alpha = 1$ .