Quantum Physics I Tough Normalization Integral Help

[Kvm90]

I am pretty sure that I'm doing this right but the integral for normalization seems impossible. Here is the question:

Normalize this wave function.
psi(x,t)=Axe^(-sqrt(km)/2h)*x^2))*e^(-i(3/2)(sqrt(k/m)*t)
for -infinity<x<+infinity where k and A are constants and m is given.

I used int(psi(x,t)psi*(x,t)dx)=1 and have arrived at the following integral, which I (nor online integrators) can't clearly do:

int(A^2*x^2*e^[(-sprt(km)/h)*x^2]=1

According to the back of the book, the answer is [sqrt(2)/pi^(1/4)][km/h^2]^(3/8). Any help would be appreciated!

 

[kreil]

This is an example of a Gaussian Integral, which is given by the following:

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

In your case if you let $\alpha = \left ( \frac{km}{h} \right )^{1/2}$, I got the following when I used Wolfram Alpha:

$$\int_{- \infty}^{\infty} x^2 e^{-ax^2} dx = \frac{\sqrt{\pi}}{2a^{3/2}}=\frac{1}{A^2}$$

Now just solve for A