Evaluation of this integral: $\int^{\infty}_{-\infty}axe^{-\frac 1 2 ax^2}\big[ax^2-3\big]dx$

Question

I am looking for some help in evaluating the following integral. I am not asking you to do it for me! \begin{equation} \int^{\infty}_{-\infty}axe^{-\frac 1 2 ax^2}\bigg[ax^2-3\bigg]dx \end{equation} I let $u=ax^2$, \begin{equation} \frac 12 \int^{\infty}_{-\infty}e^{-\frac 12 u}\bigg[u-3\bigg]du \end{equation} Yet from here I am quite stuck!

Answer 1

You may notice that $$\int^{+\infty}_{-\infty}axe^{-\frac 1 2 ax^2}\bigg[ax^2-3\bigg]dx=0$$ since the integrand is an odd function.