\begin{align*} I^2 & = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -\frac{u^2}{2} } du \cdot \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{ -\frac{v^2}{2} } dv\\ & = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{ -\frac{u^2+v^2}{2} } du dv \end{align*}