Identities for integral expressions like $\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$

Question

I have some fairly complicated integrals to work with, and sometimes to divide by each other.

$$\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$$

Are there any identities that can be used for this? I guess not, but does anyone know any?

Answer 1

I strongly doubt it. One way to think about this is that $a(x)$ is a weight function, weighting the part of the interval where it is large. There is nothing to tell you whether $f$ or $g$ is larger where $a$ is large.