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:

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


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:

\(\displaystyle \int_{\map \phi a}^{\map \phi b} \map f t \rd t\) \(=\) \(\displaystyle \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int_{a - c}^{b - c} \map f t \rd t\) \(=\) \(\displaystyle \int_a^b \map f {u - c} \paren 1 \rd u\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \map f {u - c} \rd u\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \map f {t - c} \rd t\)

$\blacksquare$