For an integrable function on prove the following identity:
Proof. First, we use the theorem on invariance of the integral under translation (Theorem 1.18 of Apostol), to write,
Then, using the expansion/contraction of the interval of integration (Theorem 1.19 of Apostol) we obtain
Why does b-a just multiply x and not a as well when using Theorem 1.19?
@Andres: the argument x is replaced, as in f(g(x)).