I need proof .$\int^{\infty}_{0} e^{-at^2} \cos(2 x t) \, dt = \frac{1}{2} \sqrt{\frac{\pi}{a}} e^{\frac{-x^2}{a}}$. $$\int^{\infty}_{0} e^{-at^2}  \cos(2 x t) \, dt = \frac{1}{2} \sqrt{\frac{\pi}{a}} e^{\frac{-x^2}{a}}$$
for Re$(a)>0$,  for all real value greater than zero.
I took this equation from the book, Abramowitz and Stegun, Equation number $7.4.6$

Hint
Consider that you look for the real part of
$$\int^{\infty}_{0} e^{-at^2} e^{2i x t} \, dt=\int^{\infty}_{0}e^{-a t^2+2 i t x}\,dt$$
Complete the square to face a well known integral.
