Imroper integral. Show that this expresions are...

Question

Show that:

$$\int_0^\infty x^2e^{-x^2} \, dx = 1/2\int_0^\infty e^{-x^2} \, dx$$

How can i prove it? Can anyone help me.

Answer 1

HINT:

Integrating by Parts,

$$\int x^2e^{-x^2}\ dx=x\int xe^{-x^2}\ dx-\int\left[\dfrac{dx}{dx}\int xe^{-x^2}\ dx\right]dx$$

For $\int xe^{-x^2}\ dx,$ set $x^2=y$