Proving an iIf a function $f: [a,b] -> \mathbb{R}$ is differentiable, then for random $c, c' \in [a,b]$ntegral property

Question

Prove. If a function $f: [a,b] -> \mathbb{R}$ is differentiable, then for random $c, c' \in [a,b]$

$ \int_{a}^{c} f(x) dx - \int_{a}^{c'} f(x) dx = \int_{c'}^{c} f(x) dx $.

I have already proven when $ a<b$ then

$\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$

Could I possibly use that to show what I want to know?

Answer 1

Hint: $(c-a)-(c'-a) = (c-c')+(c'-a)-(c'-a)$.