Dealing w/slight modification of Gaussian integral?

[Zacarias Nason]

Hey, folks. I'm doing a problem wherein I have to evaluate a slight variation of the Gaussian integral for the first time, but I'm not totally sure how to go about it; this is part of an integration by parts problem where the dv is similar to a gaussian integral:

$$\int_{-\infty}^{\infty}x^2e^{-\alpha x^2} dx, \ \text{where} \ \alpha > 0$$. My question is, how should I deal with the alpha in the exponent? I know the Gaussian integral equals
$$\sqrt{\pi}$$

But how do I deal with that alpha?

I thought I found my answer in a handout, but when I applied it, I ended up still being wrong; I attempted to integrate by parts and I got zero, the two terms cancelling each other out; what gives?

 

[Samy_A]

Hey, folks. I'm doing a problem wherein I have to evaluate a slight variation of the Gaussian integral for the first time, but I'm not totally sure how to go about it; this is part of an integration by parts problem where the dv is similar to a gaussian integral:

$$\int_{-\infty}^{\infty}x^2e^{-\alpha x^2} dx, \ \text{where} \ \alpha > 0$$. My question is, how should I deal with the alpha in the exponent? I know the Gaussian integral equals
$$\sqrt{\pi}$$

But how do I deal with that alpha?

I thought I found my answer in a handout, but when I applied it, I ended up still being wrong; I attempted to integrate by parts and I got zero, the two terms cancelling each other out; what gives?

Try a substitution to get back to the basic form: $y=\sqrt{\alpha} x$
Then integration by parts.