calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$ Could someone please help me to calculate the integral of:
$$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$
a and b both real, b>0.
I have tried integration by parts, but I can't seem to simplify it to anything useful. Essentially, I would like to arrive at something that looks like: 7.4.6 here:
textbook result

Hint:
Use the fact that
$$\int_{-\infty}^\infty e^{iat- bt^2}\,dt = \sqrt{\frac{\pi}{b}} e^{-a^2/4b} $$
which is valid for $b>0$.
To derive this formula, complete the square in the exponent and then shift the integration contour a bit.
