Proof Of Gaussian Integral Using Feynman's Technique

#Math
Proves the Gaussian Integral using Feynman's Technique
$$
\begin{align}
\int_{-\infty }^{\infty} e^{-x^{2}} , \mathrm{d}x&=2\int_{0}^{\infty} e^{-x^{2}} , \mathrm{d}x\text{ because }e^{-x^{2}}\text{ is an even function} \\&=2I
\end{align}
$$
$$
\begin{align}
I(t)=\int_{0}^{\infty} f , \mathrm{d}x
\end{align}
$$