I'm trying to find the asymptotic expansion for 
$$ F(\beta)=\int\_{-a}^{a} dx \, \frac{1}{e^{\beta g(x)}+1} \;\;\;\;\;\;\mbox{as}\;\;\;\;\;\; \beta \rightarrow +\infty.$$
where $a>0$ and $g(x)$ is a smooth, bounded and even function.

For example $g(x)=cos(x)$