# 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:

$\displaystyle \int_{a - c}^{b - c} f \left({t}\right) \ \mathrm d t = \int_a^b f \left({t - c}\right) \ \mathrm d t$

## Proof

Let $\phi \left({u}\right) = u - c$.

$\phi' \left({u}\right) = 1$
 $\displaystyle \int_{\phi \left({a}\right)}^{\phi \left({b}\right)} f \left({t}\right) \ \mathrm d t$ $=$ $\displaystyle \int_a^b f \left({\phi \left({u}\right)}\right) \phi' \left({u}\right) \ \mathrm d u$ $\displaystyle \implies \ \$ $\displaystyle \int_{a - c}^{b - c} f \left({t}\right) \ \mathrm d t$ $=$ $\displaystyle \int_a^b f \left({u - c}\right) \left({1}\right) \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle \int_a^b f \left({u - c}\right) \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle \int_a^b f \left({t - c}\right) \ \mathrm d t$

$\blacksquare$