finding a value of gamma function in terms of $\sqrt{\pi}$

Question

The Gamma function is: $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ and I have an integral: $$ \int_0^\infty x^{2} e^{-x^2}\,dx $$ I figured out that $\alpha=3$

and if $\alpha = 3 \\ \Gamma(3)= 3*2*1$

But how do I express the value of the integral in terms of $\sqrt{\pi}$? $$ \int_0^\infty x^{2} e^{-x^2}\,dx=\frac14\sqrt{\pi}\ ? $$

Answer 1

Using the substitution $u = x^2$, we get $$\int_0^\infty x^2 e^{-x^2}\,dx = \int_0^{\infty} ue^{-u} (\frac{u^{-1/2}}{2}\, du) = \frac{1}{2}\int_0^\infty u^{1/2}e^{-u}\, du = \frac{1}{2}\Gamma(3/2) = \frac{1}{4}\Gamma(1/2) = \frac{1}{4}\sqrt{\pi}$$

Answer 2

$$\Gamma \left(\frac{1}{2}\right) =\sqrt{\pi}$$ and by change of variable $x^2\to u$ should do it.