We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
(a) Show that for all integer $n \in \mathbb{N}$, we have
$$n! = \int_0^{+\infty} e^{-t} t^n \mathrm{d}t$$
(b) Using the preceding results, recover Stirling's formula giving an asymptotic equivalent of $n!$.