I'm hoping to determine the value of the following integral: $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$$
Here is a plot of the integrand as a function of $x$ with parameter $n$ varying from 0 to 10.
The integral appears to not converge. However, it is known that $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n x) \, \mathrm{d}x = 2\pi \delta(n)$$
where $\delta(x)$ is the Dirac delta function. Is it possible for the first integral to be expressed similarly using Dirac delta notation?