= Solution
For $p>0$, <integration by parts> gives the <Gamma function recurrence>:
$$
\Gamma(p+1)=\int_0^\infty x^pe^{-x}dx=[-x^pe^{-x}]_0^\infty+p\int_0^\infty x^{p-1}e^{-x}dx=p\Gamma(p).
$$
The endpoint terms vanish because $x^p\to0$ at zero and the exponential dominates the power at infinity. Also $\Gamma(1)=\int_0^\infty e^{-x}dx=1$. Iterating for nonnegative <integers> $n$ therefore yields
$$
\boxed{\Gamma(n+1)=n!.}
$$
Negative integers are outside this integral definition and factorial conclusion.
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