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

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

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

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

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