Deduce an integral I came across

[Funzies]

Hey guys, I'm trying to deduce an integral I came across whilst studying some thermodynamics, but I can't seem to evaluate it:
$$\int_0^\infty x^3e^{-x^2}dx$$
I've tried partial integration numerous times, but I can't seem to get it right. Can you help me?
I consider
$$\int_{-\infty}^\infty e^{-x^2}dx = \frac{1}{2}\sqrt{\pi}$$
as a standard integral.

 

[micromass]

Try substitution $$u=x^2$$.

 

[Funzies]

Thanks, I did try that substitution before, but now I tried it again and it came out, I must have made a mistake. Thanks!

 

[HallsofIvy]

Specically, write the integral as
$$\int_0^\infty x^2e^{-x^2}(x dx)$$
and then set $u= x^2$. You will get an integral of the form
$$\frac{1}{2}\int_0^\infty ue^{-u}du$$
that you can integrate by parts.