Q:

Calculate $\iint\limits_{\mathbb{R}^2} (x^2+y^2)^p e^{-x^2-y^2} d x d y$.

I want to calculate an integral
$$
\iint\limits_{\mathbb{R}^2} (x^2+y^2)^p e^{-x^2-y^2} d x d y,
$$
using gamma-functions.

A:

Fill in details in the following:
Use first polar coordinates , so your integral is
$$\int_0^{2\pi}\int_0^\infty r^{2p}e^{-r^2}rdr\,d\theta=2\pi\int_0^\infty r^{2p+1}e^{-r^2}dr\;\;(**)$$
and now a new substitution: $\;t:=r^2\implies dt=2r\,dr\implies dr=\frac{dt}{2\sqrt t}\;$:
$$(**)=2\pi\int_0^\infty t^{p+1/2}e^{-t}\frac{dt}{2\sqrt t}=\pi\,\Gamma(p+1)$$
...as long as $\;\Gamma(p+1)\;$ is defined , of course.

