Definite Integral of Odd Function/Corollary

Corollary to Definite Integral of Odd Function

Let $f$ be an odd 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:

$\ds \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x = 0$

Proof

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

\(=\)

\(\ds \lim_{y \mathop \to a} \int_{-y}^y \map f x \rd x\)

Definition of Improper Integral over Open Interval

\(\ds \)

\(=\)

\(\ds \lim_{y \mathop \to a} 0\)

Definite Integral of Odd Function

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$