Proof of Gaussian Integral using Feynman's Technique

Proves the Gaussian Integral using Feynman’s Technique

$$\begin{array}{rl}{\int }_{-\infty }^{\infty }{e}^{-x^2}\mathrm{d}x& =2{\int }_0^{\infty }{e}^{-x^2}\mathrm{d}x\text{ because }{e}^{-x^2}\text{ is an even function}\\ & =2I\end{array}$$ $$\begin{array}{r}I(t)={\int }_0^{\infty }f\mathrm{d}x\end{array}$$