Integral of $x^3e^{(-x^2/2)}$

Question

I have been able to calculate the integral of

$$\int^\infty_\infty x^2e^{-x^2/2}$$

and there is a lot of information online about integration with even powers of $x$.
However I have been unable to calculate:

$$\int^\infty_\infty x^3e^{-x^2/2}.$$

The closest I have come to finding a solution is
$$\int^\infty_0 x^{2k+1}e^{-x^2/2} = \frac{k!}{2}$$

Which I found here.

Any help with solving this integral would be great.

Answer 1

Do you mean $\int_{-\infty}^\infty x^3e^{-\frac{x^2}2}\,\mathrm dx$? It is $0$, since the function is an odd function and integrable (it is the product of a polynomial function with $e^{-\frac{x^2}2}$).

Answer 2

Substitute $$u=x^2$$ then we get $$\frac{1}{2}\int e^{-u/2}udu$$ and then use Integration by parts.