$f:\mathrm{R}\to \mathrm{R} , g:\mathrm{R}\to \mathrm{R}$ are continuous functions. The value of integral ${\int }_{-\pi /2}^{\pi /2} \left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx$ is 

$I={\int }_{-\pi /2}^{\pi /2} \left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx$

As  $f\left(x\right)+f(-x)$ represent even function and $g\left(x\right)-g(-x)$ represent odd function 

We know product of even and odd function is odd function

So , by property (V) $I={\int }_{-\pi /2}^{\pi /2} \left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx=0$