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I am working on the following exercise:

Let $h: [a,b] \rightarrow \mathbb{R}$ be a monotonic function and let $f: [a,b] \rightarrow \mathbb{R}$ a continuous function. Then exists a $\xi \in (a,b)$ sucht that

$$\int_a^b h(x)f(x) dx = h(a^+)\int_a^\xi f(x) dx + h(b^-) \int_\xi^b f(x) dx.$$

As I understand it this is a generalisation of the "Second Mean Value Theorem for definite integrals". The requirements that $h$ is positive and decreasing are dropped and since $f$ is continuous it is Riemann-integrable.

I do not know how to prove that. Could you help me?

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3nondatur
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    This question is a duplicate of https://math.stackexchange.com/questions/3138717/int-ab-fxgx-dx-ga-int-ax-0fx-dxgb-int-x-0bfxdx?noredirect=1&lq=1 – Angela Pretorius Nov 05 '19 at 17:33

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Without loss of generality assume that $h$ is monotonically increasing. Let $\tilde h(x)=h(x)-h(a)$. Note that $\tilde h(x)\ge 0$.

You want to prove that there exists a $\xi$ such that $\int_a^b \tilde h(x)f(x) dx = \tilde h(b) \int_\xi^b f(x) dx$. See here for the proof of this statement.

Angela Pretorius
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