calculus - Integrals involving exponential functions and the gamma function

I'm having trouble evaluating this integral

$$\int_0^\infty {e^{-ax^2}} \,dx $$

My guess is that it would evaluate into something like

$$\int_0^\infty \frac 12e^{-s}s^{\frac 12} \ldots \,dx = \frac {\Gamma\left(\frac 12\right)}{\frac{a^{\frac 12}}{2}}$$

When you do a substitution $ \sqrt{s}= \sqrt{a}x $ so that $ s = ax^2 $. I'm having trouble convincing myself though that $ \frac {d}{ds}\sqrt{s} = \left(\ldots a^\frac 12\right) $ which would satisfy the answer that I provided.

Am I doing something wrong or is my guess wrong?

Answer

If you want to use the Gamma function the substitution is $a x^2 = t$, so “$dx = \frac{1}{2\sqrt{a}}t^{-1/2}dt$".
Then the integral appears as,
$$\frac{1}{2\sqrt{a}} \int_0^\infty dt\,t^{-1/2}e^{-t} = \frac{1}{2\sqrt{a}}\Gamma(1/2)\ .$$

That's all.