TITLE: Evaluate the integral $\int^{\infty}_{0} e^{-x}x^{100}dx$
QUESTION [7 upvotes]: $$\int^{\infty}_{0} e^{-x}x^{100}dx$$

I am sure is something here I can not see, else it is integration by parts 100 times.

REPLY [2 votes]: This is the Gamma function of $101$. Indeed by definition:
$$\Gamma[x] = \int_0^{+\infty} t^{x-1}\ e^{-t}\ \text{d}t$$
And so
$$\int_0^{+\infty} t^{100}\ e^{-t}\ \text{d}t = \Gamma[101] = 100!$$
About the integration
This integration can be done in several ways. By parts is surely the most intuitive one, even if it may be tedious (although you need just two-three passages to guess the whole behavior of the integration).
Otherwise, we can do it with the Feynman trick as shown below/above.