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?

(b-a)integral_0^1 f(a+(b-a)x)dx