This question came up in the real analysis 1 exam I wrote today (I hope my memory is accurate!) and I had no clue how to approach this question, so any tips would be appreciated.
Let $f:\mathbb{R} \to \mathbb{R}$ be differentiable and $f'(x)$ continuous and $g(t) = \int\limits_{a}^{b} f'(x+t) dx$.
a) Explain why $g$ is well-defined
b) Prove $\int\limits_{a}^{b} g(x) dx = \int\limits_{a+b}^{2b} f(x) dx - \int\limits_{2a}^{a+b} f(x) dx$
For b) I tried to argue by showing that $$\int\limits_{a+b}^{2b} f(x) dx - \int\limits_{2a}^{a+b} f(x) dx = \int_{a}^{b} f(2x) dx - 2\int_{a}^{b} f(x) dx$$ but that nowhere.
Hint
Using a linear change of variables $ x \rightarrow x+t$: $$g(t)=\int_a^b f'(x+t) dt= \int_{a+t}^{b+t} f'(x) dx=f(b+t)-f(a+t)$$ so: $$\int_a^b g(t) dt = \int_a^b f(b+t) dt - \int_a^b f(a+t) dt$$ and you can obtain the result by linear changes of variables in each integral.