I can compute the following integral very easily ($a$ and $b$ are real and positive):
$$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = e^{-\frac{a^2b}{4}}$$
Now, I want to compute the following integral in terms of $a$ and $b$:
$$\int_{-\infty}^\infty |\cos(ax)|\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx =\text{ ?}$$