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