Definite Integral of Even Function/Corollary

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:

$\ds \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x = 2 \int_0^{\mathop \to a} \map f x \rd x$

$\ds \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x$

$=$

$\ds \lim_{y \to a} \int_{-y}^y \map f x \rd x$

$\ds $

$=$

$\ds \lim_{y \mathop \to a} 2 \int_0^y \map f x \rd x$

$\ds $

$=$

$\ds 2 \lim_{y \mathop \to a} \int_0^y \map f x \rd x$

$\ds $

$=$

$\ds 2 \int_0^{\mathop \to a} \map f x \rd x$

$\blacksquare$