How many iterations of integration by parts?

Calculus Level 3

$40320 = \int_{0}^{\infty} x^{\mathscr{E}} e^{-x} \, dx$

Find $\mathscr{E}$ that makes the above equation true.

1 solution

Rishabh Jain
Jun 16, 2016

$\underbrace{40320}_{8!} = \int_{0}^{\infty} x^{\mathscr{E}} e^{-x} \, dx =\Gamma(\mathscr{E}+1)=\mathscr E!$

$\Large \implies \mathscr E=\boxed{\color{#007fff}{8}}$