Calculate the integral of x^2 e^(-x^2) from -infinity to + infinity

[stunner5000pt]

$$\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{\infty} x^2 e^{\frac{-x^2}{2 \sigma^2}} dx$$
I think just slving this would be fine too

$$\int_{-\infty}^{\infty} x^2 e^{-x^2} dx$$

what is the trick to solving this?/

Cnat integrate by parts because that would yeild Erf function which i have no been taught

I was told that there was some trick to differentiate both sides by sigma... but I am not relly sure...?

 

[StatusX]

You know:

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

Differentiate both sides with respect to sigma.