Use integration by parts to prove $ \int^\infty_0 x^ne^{-x} dx=n!$

Question

$ \int^\infty_0 (x^n)(e^{-x}) dx $ and show that it is equal to $(n!)$ ? I know that if you differentiate $x^n$ infinitely you get $n!$ but I don't know how to prove?

Answer 1

1. Show by parts that if $I_m = \int_0^{\infty} x^m e^{-x} \mathrm{d}x$, then $$I_m = m I_{m-1}$$
2. Show that $I_0 = 1$.
3. Conclude inductively that $I_n = n!$ for all $n$.