Question

evaluate the integral of general odd function from negative infinity to infinity?

Answer 1

\[I=\int\limits_{-\infty}^{\infty}f \left( x \right)~dx,where~f \left( x \right)~is~an~odd~function.\]
put x=-y,
dx=-dy
\[when~x \rightarrow \infty,y \rightarrow-\infty ,when~x \rightarrow-\infty,y \rightarrow \infty \]
\[I=\int\limits_{\infty}^{-\infty}f \left( -y \right)\left( -dy \right)=-\int\limits_{\infty}^{-\infty}f \left( -y \right)dy\]
\[I=\int\limits_{-\infty}^{\infty}f \left( -y \right)dy=\int\limits_{-\infty}^{\infty}f \left( -x \right)dx=-\int\limits_{-\infty}^{\infty}f \left( x \right)dx=-I\]
I+I=0
2I=0
I=0

Answer 2

if f(x)is an odd function,then
f(-x )=-f(x )