Integration by Substitution/Corollary

Corollary to Integration by Substitution
Let $f : \R \to \R$ be a real function.

Let $f$ be integrable.

Let $a$, $b$, and $c$ be real numbers.

Then:


 * $\displaystyle \int_{a - c}^{b - c} \map f t \rd t = \int_a^b \map f {t - c} \rd t$

Proof
Let $\map \phi u = u - c$.

By Sum Rule for Derivatives, Derivative of Identity Function, and Derivative of Constant, we have:


 * $\map {\phi'} u = 1$

By Integration by Substitution: