Definite Integral of Even Function/Corollary
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Corollary to Definite Integral of Even Function
Let $f$ be an even function with a primitive on the open interval $\openint {-a} a$, where $a > 0$.
Then the improper integral of $f$ on $\openint {-a} a$ is:
- $\ds \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x = 2 \int_0^{\mathop \to a} \map f x \rd x$
Proof
\(\ds \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x\) | \(=\) | \(\ds \lim_{y \to a} \int_{-y}^y \map f x \rd x\) | Definition of Improper Integral over Open Interval | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{y \mathop \to a} 2 \int_0^y \map f x \rd x\) | Definite Integral of Even Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \lim_{y \mathop \to a} \int_0^y \map f x \rd x\) | Multiple Rule for Limits of Real Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \int_0^{\mathop \to a} \map f x \rd x\) | Definition of Improper Integral over Open Interval |
$\blacksquare$