Question

Let f : R $\to$ R and g : R $\to$ R be continuous functions, then the value of the integral ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx=$ [IIT 1990; DCE 2000; MP PET 2001]

• A.
• B. 1
• C. -1

Answer 1

$Leth\left(x\right)=\left\{f\right(x)+f(-x\left)\right\}\left\{g\right(x)-g(-x\left)\right\}$
$h(-x)=\left\{f\right(-x)+f(x\left)\right\}\left\{g\right(-x)-g(x\left)\right\}$
$=-\left\{f\right(-x)+f(x\left)\right\}\left\{g\right(x)-g(-x\left)\right\}=-h\left(x\right)$
$Therefore,{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}h\left(x\right)dx=0$