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

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.

• Substitute $u=x^2/2$, and integrate by parts – Jakobian Oct 31 '18 at 15:42
• @Jakobian got it, thank you I was thinking too much in terms of Gaussian integrals and missed the obvious. – Matthew Oct 31 '18 at 15:45

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 $$e^{-\frac{x^2}2}$$).
• Just because it's odd, isn't enough to say that the integral is $0$. The integral doesn't have to exists – Jakobian Oct 31 '18 at 15:45
Substitute $$u=x^2$$ then we get $$\frac{1}{2}\int e^{-u/2}udu$$ and then use Integration by parts.