Messy Gaussian Integral I am trying to understand how to better perform the following integral.
$$\int^{\infty}_{0} x^4 e^{\frac{-x^2}{\beta^2}}\mathrm{d}x$$
I've done a little research and found that $e^{-x^2}$ doesn't integrate easily, for it is the Gaussian integral. Many sources are pointing me to use polar coordinates. Is that the best way to go about this? How can I solve this integral?

$$\int_{0}^{\infty} x^p e^{-ax}=\frac{\Gamma(p+1)}{a^{p+1}}.$$
So $$I=\int_{0}^{\infty}  x^4 e^{-x^2/b^2} dx=\int_{0}^{\infty} \frac{b^5}{2} t^{3/2} e^{-t} dt= \frac{b^5}{2}\Gamma(5/2)=\frac{3 \sqrt{\pi} b^5}{8}. $$
