A quick question concerning error function

Question

Why $\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty}e^{tx}e^{-x^{2}/2}dx$

equals to $e^{t^{2}/2}$ ?

I know it is error function. but I just do not have any basic knowledge about error function and do not know how to derive it.

Answer 1

$e^{tx-x^{2}/2}=e^{-(x-t)^{2}/2} e^{t^{2}}$, Make the substitution $y=x-t$ to evaluate the integral.