Evaluate $\int_{-\infty}^\infty \cos(\omega t)e^{-t^2} dt$

Question

I'd like to show the equality

$$ \int_{-\infty}^\infty \cos(\omega t)e^{-t^2} dt = \sqrt{\pi} e^{-\omega^2 / 4} $$

I tried to solve this by using the contour integral for the upper half disk but since the integrad involves two exponentials, it seems hard to prove in this way. Is there any other good way to solve this?