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
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 ,
Then we have
Thus, we indeed have