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:
$\displaystyle∫_{−∞}^\infty \exp(−^2 )\ =\sqrt{}$
$\displaystyle∫_{−∞}^\infty \exp(−a^2 )\ =\sqrt{a}\ ??$
$$I=\int_{-\infty}^{\infty} e^{-ax^2} dx$$ Let $x\sqrt{a}=t \implies dx=\frac{dt}{\sqrt{a}}$, then $$I=\int_{-\infty}^{\infty} e^{-t^2} \frac{dt}{\sqrt{a}}=\frac{\sqrt{\pi}}{\sqrt{a}} .$$