# Definite Integration of Odd and Even Functions – Property 1

## Definite Integration of Odd and Even Functions – Property 1

Property 1:

$$\int\limits_{-a}^{a}{f\left( x \right).dx}$$=\left\{ \begin{align} & 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ f(x)\ \ is\ \ odd\ \ i.e,\ f(-x)=-f(x) \\ & 2\int\limits_{0}^{a}{f(x)dx,\ \ \ if\ \ f(x)\ \ is\ \ even\ \ i.e,\ f(-x)=f(x)} \\ \end{align} \right.

Proof:

$$\int\limits_{-a}^{a}{f\left( x \right).dx}=\int\limits_{-a}^{0}{f\left( x \right).dx}+\int\limits_{0}^{a}{f\left( x \right).dx}$$

Put x = -t in first term on R.H.S.

Differentiation with respect to ‘x’

dx = – dt

when x = -a

x = -t

-a = -t

a = t and

x = -t

put x = 0

0 = -t

t = 0

$$\int\limits_{-a}^{a}{f\left( x \right).dx}=\int\limits_{a}^{0}{f\left( -t \right).\left( -dt \right)}+\int\limits_{0}^{a}{f\left( x \right).dx}$$,

$$=\int\limits_{0}^{a}{f\left( -t \right).\left( dt \right)}+\int\limits_{0}^{a}{f\left( x \right).dx}$$,

$$=\int\limits_{0}^{a}{f\left( -x \right).\left( dx \right)}+\int\limits_{0}^{a}{f\left( x \right).dx}$$,

$$=-\int\limits_{0}^{a}{f\left( x \right).\left( dx \right)}+\int\limits_{0}^{a}{f\left( x \right).dx},$$  if f(x) is odd

$$=\int\limits_{0}^{a}{f\left( x \right).\left( dx \right)}+\int\limits_{0}^{a}{f\left( x \right).dx},$$ if f(x) is even

$$=2\int\limits_{0}^{a}{f\left( x \right).\left( dx \right)}+\int\limits_{0}^{a}{f\left( x \right).dx},$$

$$\int\limits_{-a}^{a}{f\left( x \right).dx}$$ x=\left\{ \begin{align} & 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ f(x)\ \ is\ \ odd\ \ i.e,\ f(-x)=-f(x) \\ & 2\int\limits_{0}^{a}{f(x)dx,\ \ \ if\ \ f(x)\ \ is\ \ even\ \ i.e,\ f(-x)=f(x)} \\ \end{align} \right.