Integration by Substitution/Corollary

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

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

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


By Integration by Substitution:

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

$\blacksquare$