I am hoping to find an explicit evaluation of the integral $$ \int_0^\infty \cos(p(x)) dx $$ and similarly $$ \int_0^\infty \sin(p(x)) dx $$ where $p(x)$ is a polynomial of degree greater than or equal to 2. What I am hoping for is an explicit formula in terms of the coefficients of $p(x)$.
A particular special case is the limit of the Fresnel integrals and I was hoping there is a generalization of that. Perhaps there is not some simple evaluation in terms of known constants for a general polynomial. Here is a question that I found related to this that solves a certain special case.