Gaussian Integral: How to Solve for x^4 Term?

[nayfie]

Homework Statement

I'm having difficulty solving the following integral.

$\int_{-\infty}^{\infty} x^{4}e^{-2\alpha x^{2}} \text{d}x$

Homework Equations

$\int_{-\infty}^{\infty} e^{-\alpha x^{2}} \text{d}x = \sqrt{\frac{\pi}{\alpha}}$

$\int_{-\infty}^{\infty} x^{2}e^{-\alpha x^{2}} \text{d}x = \frac{\sqrt\pi}{2\alpha^{\frac{3}{2}}}$

The Attempt at a Solution

I solved a very similar integral, however this one was much easier as I could use substitution quite easily. (In this case I let $u = x^{2}$ and $\text{d}u = 2x\text{d}x$).

$\int_{-\infty}^{\infty} x^{3}e^{-2\alpha x^{2}} \text{d}x$

(It turns out that the integral above is zero.)

The only difference in this new question is a factor of x, yet I have no idea how to approach it. Ideally I would want to find a way to manipulate this integral to be of a form that I know the solution (e.g the ones I've listed above).

Just need a point in the right direction. Any help would be greatly appreciated!

 

[hamsterman]

Are you familiar with this? If you integrate what you have by alpha, you'll get the second relevant equation.

 

[nayfie]

I haven't been taught that yet. Is that the only way to solve this integral? I've been reading the related articles but can't work out how to apply it to this question.

Unfortunately it seems my physics course is ahead of my maths course.

 

[Office_Shredder]

Take the second relevant equation and do integration by parts - integrate the x² and differentiate the e^{-2a x^2} to get an x⁴ e^{-2ax^2} integral. Then solve for this new integral in terms of things that you know

 

[nayfie]

Took your advice and the answer popped out straight away. How did I not think of this?

Thanks mate :)