Question

If $f:R\to R$ and $g:R\to R$ are continuous sunction, then the value of ${\int }_{-\pi /2}^{\pi /2}\left[(f\left(x\right)+f(-x))(g\left(x\right)-g(-x))\right]dx$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $\pi$

Answer 1

The correct option is A $0$

Two continuous function are $f:R\to R$ and $g:R\to R$.

Let $F\left(x\right)=|f\left(x\right)+f(-x)||g\left(x\right)+g(-x)|$ ...(1)

and $F(-x)=|f(-x)+f\left(x\right)||g(-x)+g\left(x\right)|$ ...(2)

Adding equation (1) and (2), we get

$F\left(x\right)+F(-x)=|2f\left(x\right)+2f(-x)||g\left(x\right)-g(-x)+g(-x)-g\left(x\right)|\Rightarrow F\left(x\right)+F(-x)=0\Rightarrow F\left(x\right)=F(-x)$

$\Rightarrow F\left(x\right)$ is an odd function.

Therefore ${\int }_{-\pi /2}^{\pi /2}\left[(f\left(x\right)+f(-x))(g\left(x\right)+g(-x))\right]dx=0$