We admit the identity
$$\int _ { - \infty } ^ { + \infty } \exp \left( - x ^ { 2 } \right) \mathrm { d } x = \sqrt { \pi }$$
(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 $n !$.