# What is this result called: $\int_a^b f(x)g(x)dx=f(c)\int_a^b g(x)dx$?

there's $c\in(a,b)$ such that

$$\int_a^b f(x)g(x)dx=f(c)\int_a^b g(x)dx$$

Can anyone help me identify this rule? Thank you! =)

Mean value theorem for integration. In case you want to know how this name came about: taking $g=1$, we see that there is $c$ with $$f(c)=\frac{1}{b-a}\int_a^b f \ dx,$$ the right hand side of which is called the integral mean of $f$ over $[a,b]$. (These integrals are fairly important, e.g. because it appears in the Lebesgue differentation theorem; in fact so important that there is even a standard notation. Well it should be without the absolute value bars, but somehow no code seems to work here.)