I am asked to simplify/calculate the following integral: $$\frac{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] \right)|u(q)|^2}{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] \right)}$$ Where the $q$ is some vector and $u(q)$ is the scalar displacement. Now, I was wondering how to solve this integral and I thought of this: if the integral inside the exponential was instead a discrete sum, I could write it as a product, because $e^{a+b}=e^a e^b$.
My question is two fold: is this indeed the way to solve this integral, and if so, how does this work with integrals?