Note that

and use the weighted mean value theorem (Theorem 3.16 in Apostol) to prove:

Recall the weighted mean value theorem:

For functions and continuous on , if never changes sign in then there exists such that

* Proof. * Let

Then,

for some . Since is strictly increasing on , we have implies . Thus,

We also know

So, putting these together,

How can you merge f(c) and integral of g(x)?

It seems safer to do it as in the proof of 3.16.

Sorry, the way you proved it seems valid and probably the intended way, because it uses the required theorem.