How do I integrate an exponential with a higher power?

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To integrate the function ∫x^3e^{-\alpha x^2}dx, a useful approach is to substitute u = x^2, simplifying the integral to involve e^{-au}. This leads to an integral that can be solved using integration by parts. The integration by parts technique is necessary due to the presence of the x^3 term, which complicates direct integration. Many participants in the discussion confirm that the substitution and subsequent integration by parts is the correct method to tackle the problem. Overall, the integration process requires practice and familiarity with these techniques.
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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!
 
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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.
 
\int x^3e^{-\alpha x^2}dx= \int x^2 e^{-\alpha x^2} (xdx)

Now, as others have said, let u= x2.
 
Oh so I was going right.

I hate integration by parts.
 

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