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 $\left({-a \,.\,.\, a}\right)$, where $a > 0$.


Then the improper integral of $f$ on $\left({-a \,.\,.\, a}\right)$ is:

$\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \rd x = 2 \int_0^{\mathop \to a} f \left({x}\right) \rd x$


Proof

\(\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \rd x\) \(=\) \(\displaystyle \lim_{y \to a} \int_{-y}^y f \left({x}\right) \rd x\) Definition of Improper Integral over Open Interval
\(\displaystyle \) \(=\) \(\displaystyle \lim_{y\to a} 2 \int_0^y f \left({x}\right) \rd x\) Definite Integral of Even Function
\(\displaystyle \) \(=\) \(\displaystyle 2 \lim_{y\to a} \int_0^y f \left({x}\right) \rd x\) Multiple Rule for Limits of Functions
\(\displaystyle \) \(=\) \(\displaystyle 2 \int_0^{\mathop \to a} f \left({x}\right) \rd x\) Definition of Improper Integral over Open Interval

$\blacksquare$