$f,g$ cont. functions on $[a,b]$ and $g$ is non-negative. then $\exists c \in [a,b]$ such that $\int_{a}^{b} f(x)g(x) dx = f(c) \int_{a}^{b} g(x) dx.$
If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...