calculus - (Parameter Choice) Using Differentiation under integral for $e^{-x^2}$

I have the integral:

$$\int_{-\infty}^\infty e^{-x^2} \, dx$$

And I’d like to solve it using differentiation under the integral sign. I understand that I must convert $e^{-x^2}$ to $e^{-x^2}g(x,t)$, where $g(x,t)$ is just some term involving a new parameter $t$. I tried multiplying by $e^{-2xt}$, which doesn’t really seem very logical, however, I just do not know what to put. How do I find the appropriate term to make the whole process work?