Is There an Easier Way to Solve this Definite Integral?

[simo]

Homework Statement

solve the integral: ∫_(-∞)^∞▒〖x^2 e^(-λ(x-a)^2 ) 〗 dx
where λ and a are positive real constants

The Attempt at a Solution

I tried integration by parts with and without y-substitution but neither worked for me.

Without substitution, I set up the integral to look like:
∫_(-∞)^∞▒〖xe^(-λx^2 )•xe^λa(2x-a) 〗 dx

u=xe^λa(2x-a) and dv=xe^(-λx^2 ) dx

after doing this a few times I realized it wouldn't work.

For y-substitution I used y = x-a. ∫_(-∞)^∞▒〖(y+a)^2 e^(-λ(y)^2 ) 〗
I then tried to integrate this by parts with u=(y+a)^2 and dv=e^(-λy^2 )

 

[CompuChip]

I assume you want to solve
$$\int_{-\infty}^{\infty} x^2 e^{-\lambda (x - a)^2 } \, dx$$

In that case, try differentiation of an ordinary Gaussian integral w.r.t $\lambda$ (twice).

 

[simo]

Yea, I didn't have it in the right form. It's for a physics class, so the books says to use a table to help. I think I will try to solve it out anyway. Thanks for the help.