TITLE: Integration of gaussian times absolute value of cosine
QUESTION [5 upvotes]: Is there a way to compute/estimate the following integral?
$\int_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$
where $c$ is a real constant. I would like to know if it is of order $e^{-c^2/4}$ like the integral without absolute value. Or is there a sharp upperbound?
Thank you very much in advance.

REPLY [2 votes]: Just so we can actually see what we're dealing with:
Let $\displaystyle I_c=\int_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$.
This is a plot of $I_c$ versus $c$.

Now, this is a plot of $\displaystyle \frac{I_c}{c}$ for $c \in (0,5]$.

Credits to MATLAB, as always.