Find the area under a Gaussian

Question

I have a Gaussian function defined by: $h_{\alpha}(x)=\exp(-\alpha x^2)$

How do I find the area under the graph for an arbitrary value of $\alpha$

My working:

$\displaystyle∫_{−∞}^\infty \exp⁡(−^2 )\ =\sqrt{}$

$\displaystyle∫_{−∞}^\infty \exp⁡(−a^2 )\ =\sqrt{a}\ ??$

Answer 1

$$I=\int_{-\infty}^{\infty} e^{-ax^2} dx$$ Let $x\sqrt{a}=t \implies dx=\frac{dt}{\sqrt{a}}$, then $$I=\int_{-\infty}^{\infty} e^{-t^2} \frac{dt}{\sqrt{a}}=\frac{\sqrt{\pi}}{\sqrt{a}} .$$