Question

Let $f:R\to R$ and $g:R\to R$ be continuous functions. Then, the value of the integral
$\underset{-\pi /2}{\overset{\pi /2}{\int }}[f(x)+f(-x)][g(x)-g(-x)]dx$ is

• A. $\pi$
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (D)

Let $I=\underset{-\pi /2}{\overset{\pi /2}{\int }}[f(x)+f(-x)][g(x)-g(-x)]dx$
Let $\varphi (x)=[f(x)+f(-x)][g(x)-g(-x)]$
$\Rightarrow \varphi (-x)=[f(-x)+f(x)][g(-x)-g(x)]$
$\Rightarrow \varphi (-x)=-\varphi (x)$
$\Rightarrow \varphi (x)$ is an odd function.
$\therefore \underset{-\pi /2}{\overset{\pi /2}{\int }}\varphi (x)dx=0$