Serious second mean value theorem for integration

[jostpuur]

The claim:

If $f:[a,b]\to\mathbb{R}$ is integrable, and $\phi:[a,b]\to\mathbb{R}$ is monotonic (hence continuous almost everywhere), then there exists $\xi\in ]a,b[$ such that
$$\int\limits_a^b f(x)\phi(x)dx \;=\; \big(\lim_{x\to a^+}\phi(x)\big) \int\limits_a^{\xi} f(x)dx \;+\; \big(\lim_{x\to b^-}\phi(x)\big) \int\limits_{\xi}^b f(x)dx$$

Who knows how to prove that?

Or who knows a serious book on calculus, that would cover this? Or a publication that could be found in university libraries?

I found the claim from Wikipedia: http://en.wikipedia.org/wiki/Mean_value_theorem But no proof.

I don't remember where, but somewhere some years ago I found a website, that gave a proof for a weaker formulation of this theorem. It goes like this:

If $f:[a,b]\to\mathbb{R}$ is continuous, and $\phi:[a,b]\to\mathbb{R}$ is differentiable such that $\phi'\geq 0$, then there exists $\xi\in [a,b]$ such that
$$\int\limits_a^b f(x)\phi(x)dx \;=\; \phi(a) \int\limits_a^{\xi} f(x)dx \;+\; \phi(b) \int\limits_{\xi}^b f(x)dx$$
This can be proven by first substituting
$$f(x) = D_x\int\limits_a^x f(u)du$$
then integrating by parts, and then using the first mean value theorem.

 

[fresh_42]

You first prove $\displaystyle{\int_a^b }f(x)\phi(x)\,dx = f(\xi) \displaystyle{\int_a^b}\phi(x)\,dx$ and apply partial integration and the fundamental theorem of calculus.