dear community
I have a problem with integral estimation and hope anyone could help.
The numerical calculation showed me, that integral $$\int_{-\infty}^{+\infty} \frac{\cos{ax}}{I_0(x)}dx$$ (where the $I_0$ is modified Bessel function) depends on $a$ when $a\gg 1$ as $\exp(\gamma a +\beta)$, where $\gamma\simeq -2.4$, $\beta \simeq 2.5$.
I guess, that such a nice dependency comes from some elegant mathematics connected to rapidly oscillating of the cosine function. But I can't understand how to prove it and how to calculate parameters of the exponent analytically.
I tried to find such integral in books with integral tables, but failed too.
in the graphic I plot both constituents of the underintegral-function.
Maybe there is some common method to solve problems like this?
Hint: the smallest absolute value of a zero of $I_0(z)$ is about $2.4048$.