Let $f,g:[a,b]\rightarrow\mathbb{R}$ be continuous and $g(x)\neq0$. Then there is a $\xi\in(a,b)$ s.t. $\int_a^b f(x)g(x)dx=f(\xi) \int_a^b g(x)dx.$

Question

I'm following along a calculus lecture right now and the following theorem was introduced. However, no proof or even sketch of one was given and I am wondering if anybody knows where I could look at one to help me understand this.

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Let $f,g:[a,b]\rightarrow\mathbb{R}$ be continuous with $g(x)\neq0$. Then there exists a $\xi\in(a,b)$ such that $$\int_a^b f(x)g(x)dx=f(\xi) \int_a^b g(x)dx.$$

The proof is left to the reader.

There was furthermore a corollary that stated that this does not hold for complex functions, with again, no proof given.

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I think I would try to tackle the first part making use of the fundamental theorem of calculus and the mean value theorem. Although I'm quite lost on how to do that precisely. The second part has probably to do with Euler's formula, although again I wouldn't know how to formulate a proof.

Could anybody point me where to read up on this theorem? Also apologies for potential mistranslations, my native language isn't English.

Answer 1

That is the mean value theorem for integral. You can check it on >>wikipedia<<.

You will find there a proof.