Are you as smart as Gauss?

$\begin{aligned} I & = \int_{-\infty}^\infty e^{-\frac 12 ax^2} dx & \small \color{#3D99F6} \text{Let }u = \sqrt {\frac a2} x \implies du = \sqrt{\frac a2} dx \\ & = \int_{-\infty}^\infty \sqrt{\frac 2a} e^{-u^2} du & \small \color{#3D99F6} \text{Since the integrand is even,} \\ & = 2 \int_0^\infty \sqrt{\frac 2a} e^{-u^2} du & \small \color{#3D99F6} \text{Error function erf }(z) = \frac 2{\sqrt \pi} \int_0 ^z e^{- t^2} dt \\ & = \sqrt {\frac {2\pi}a} \text{erf }(\infty) & \small \color{#3D99F6} \text{and erf }(\infty) = 1 \\ & = \boxed{\sqrt {\dfrac {2\pi}a}} \end{aligned}$

Reference: Error function $\color{#3D99F6}\text{erf }(\cdot)$