For every we have
So by the fundemental theorem of calculus we have for any interval in
How can I proof this using the inverse function theorem?
"Inverse function theorem"? What is that and what does inverse functions have to do with anything related to this problem? The value of that integral is that because of the Fundamental Theorem of Integral Calculus and because in every closed interval , as you wrote...there's nothing else to it
The inverse function theorem says that if f is continuously differentiable en has derrivative at a which is not equal to 0 then f is invertible in a neighborhood of a. The inverse is then also continuously differentiable.
So the answer is like this i think;