Change of Limits of Integration

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

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


 * $\phi' \left({u}\right) = 1$

By Integration by Substitution: