TITLE: How to evaluate $\int_{-\infty}^{\infty}e^{x-n\sinh^2 x}\ dx$
QUESTION [5 upvotes]: This is what I tried:
$$\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}\dfrac{(x-n\sinh^2 x)^k}{k!}\ dx$$
$$\sum_{k=0}^{\infty}\frac1{k!}\int_{-\infty}^{\infty}\sum_{r=0}^{k}\binom{k}{r}x^r(-n\sinh^2 x)^{k-r}\ dx$$
After this I have no idea how to proceed

REPLY [17 votes]: $$\ldots=\int_0^\infty(e^x+e^{-x})e^{-n\sinh^2 x}\,dx\ \underset{\sinh x=y}{\phantom{\big[}=\phantom{\big]}}\ 2\int_0^\infty e^{-ny^2}\,dy=\sqrt{\pi/n}.$$