That's the definition of the Gamma function...
$\displaystyle \Gamma (a)=\int_0^{\infty} t^{a-1} e^{-t} dt$.
for a>0. The first thing you learn about this function is that
$\displaystyle \Gamma (a)=(a-1)\Gamma (a-1)$ and it's easy to show that
$\displaystyle \Gamma (1)=1$. Thus $\displaystyle \Gamma (2)=1$...., whence $\displaystyle \Gamma (n)=(n-1)!$ where n is an integer.