How do I integrate an exponential with a higher power?

[Brewer]

Homework Statement

I'm being dead thick, but I can't remember how to integrate an exponential function.
$$\int x^3e^{-\alpha x^2}dx$$

Homework Equations

The Attempt at a Solution

I reckon that this shouldn't be too complex, but I've totally forgotten how to go about this question. The $$x^3$$ term means that it can't be integrated like I would normally do (i.e. $$\int ke^x = ke^{\frac{x}{k}}$$) can it? I've also tried substituting $$u=x^2$$ into it and following that through, but it doesn't seem to get anywhere.

Is this really simple, and my recent lack of practice with the techniques is just failing me, or there something more complicated about it?

Please assist, this is doing my head in!

 

[AlephZero]

Substituting u = x^2 is a good first move.

You will get a simpler integral involving e^{-au} which you can solve by integration by parts.

 

[HallsofIvy]

$$\int x^3e^{-\alpha x^2}dx= \int x^2 e^{-\alpha x^2} (xdx)$$

Now, as others have said, let u= x².

 

[Brewer]

Oh so I was going right.

I hate integration by parts.