Use the method of substitution to prove invariance under translation (Theorem 1.18 on page 81 of Apostol) and to prove expansion or contraction of the interval of integration (Theorem 1.19 on page 81 of Apostol).

** Theorem: (Invariance Under Translation)** * For a function f integrable on an interval [a,b] and for every we have *

* Proof. * If is a primitive of , then

Let

So,

Hence, we indeed have

** Theorem: (Expansion or Contraction of the Interval of Integration) ** * For a function f integrable on an interval [a,b] and for every with , *

* Proof. * Let

Then we have

Thus, we indeed have