Question

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

• A. $ \pi $
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (D)

Let $I=\int\limits_{-\pi / 2}^{\pi / 2}[f(x)+f(-x)][g(x)-g(-x)] d x$
Let $ \phi(x)=[f(x)+f(-x)][g(x)-g(-x)]$
$\Rightarrow \phi(-x)=[f(-x)+f(x)][g(-x)-g(x)]$
$\Rightarrow \phi(-x)=-\phi(x)$
$\Rightarrow \phi(x)$ is an odd function.
$\therefore \int\limits_{-\pi / 2}^{\pi / 2} \phi(x) d x=0$