Show that $\displaystyle \int_{-\infty}^\infty x^{2n}e^{-x^2}dx=(2n)!\frac{\sqrt\pi}{4^n n!}$ by differentiating the equation $\displaystyle \int_{-\infty}^\infty e^{-tx^2} dx=\sqrt{\frac{\pi}{t}}$.
This is what I proved so far:
By differentiating the given equation I have $\displaystyle \int_{-\infty}^\infty x^2e^{-tx^2}dx=\frac{\sqrt\pi}{2} t^{-\frac{3}{2}}$. So, when $t=1$, $\displaystyle \int_{-\infty}^\infty x^2e^{-x^2}dx=\frac{\sqrt\pi}{2} $. Now I used mathematical induction for the expression I have to prove. I got it for $n=1$. But stucked in proving the rest. Can anyone please help me?
Hint: what appends if you write the $n$th derivative?