Is there a generalization of second Mean Value Theorem of Integrals for non-monotonic class of functions?

The Second Mean value theorem of integrals states that:

If $f,g:[a,b]\to\mathbb{R}$ are integrable functions such that $g$ is monotone, then there exists $x_0\in [a,b]$ such that $\displaystyle\int_a^b f(x)g(x)\ dx=g(a) \cdot \displaystyle\int_a^{x_0}f(x)\ dx+g(b) \cdot \int_{x_0}^bf(x)dx$.

Is there a generalization or similar result like this for non-monotonic class of functions?