Definite Integration of Odd and Even Functions – Property II

Definite Integration of Odd and Even Functions – Property II

If f(x) is an odd function, then \(\phi \left( x \right)=\int\limits_{a}^{x}{f\left( t \right)dt}\), is an even function.

Proof: \(\phi \left( x \right)=\int\limits_{a}^{x}{f\left( t \right)dt}\),

\(\phi \left( -x \right)=\int\limits_{a}^{-x}{f\left( t \right)dt}\),

Let t = -y

dt = – dy

\(\phi \left( -x \right)=\int\limits_{-a}^{x}{f\left( -y \right)\left( -dy \right)}\),

\(=\int\limits_{-a}^{x}{f\left( y \right)dy}\),

As given f is an odd function

\(=\int\limits_{-a}^{a}{f\left( y \right)dy}+\int\limits_{a}^{x}{f\left( y \right)dy}\),

\(\int\limits_{-a}^{a}{f\left( y \right)dy}\) = 0

\(=0+\int\limits_{a}^{x}{f\left( y \right)dy}\),

\(\int\limits_{a}^{x}{f\left( y \right)dy}=\phi \left( x \right)\),

Hence, \(\phi \left( x \right)\), is an even function.

Example: Evaluate  \(\int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\log \left( \frac{a-\sin \theta }{a+\sin \theta } \right).\ }d\theta ,\ a>0\),

Solution: 

let us consider \(f\left( \theta \right)=\log \left( \frac{a-\sin \theta }{a+\sin \theta } \right)\),

\(f\left( -\theta \right)=\log \left( \frac{a-\sin \left( -\theta \right)}{a+\sin \left( -\theta \right)} \right)\),

\(f\left( -\theta \right)=\log \left( \frac{a+\sin \theta }{a-\sin \theta } \right)\),

\(f\left( -\theta \right)=-\log \left( \frac{a-\sin \theta }{a+\sin \theta } \right)\),

\(f\left( -\theta \right)=-f\left( \theta \right)\).