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I'm having trouble finding a certain generalization of the mean value theorem for integrals. Ithinkmy conjecture is true, but I haven't been able to prove it - so maybe it isn't.

Is the following true?

If [itex] F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n} [/itex] is a continuous function

and [itex]x: I \subset \mathbb{R} \rightarrow V \subset \mathbb{R}^{n}[/itex] is a continuous function

then [itex]\exists t^* \in [t_1,t_2][/itex] such that

[itex]\int_{t_1}^{t_2} F(x(t),t)\,dt = F(x(t^*),t^*)(t_2-t_1)[/itex]

I can see that it holds for each of the component functions of ##F##, but I'm not sure about the whole thing.

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# Generalization of Mean Value Theorem for Integrals Needed

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