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:
- $\ds \int_{a - c}^{b - c} \map f t \rd t = \int_a^b \map f {t - c} \rd t$
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:
\(\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t\) | \(=\) | \(\ds \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_{a - c}^{b - c} \map f t \rd t\) | \(=\) | \(\ds \int_a^b \map f {u - c} \paren 1 \rd u\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map f {u - c} \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map f {t - c} \rd t\) |
$\blacksquare$