Definite Integration Ques 33

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

(1990, 2M)

(a) $\pi$

(b) 1

(c) -1

(d) 0

Show Answer

Solution:

  1. Let $I=\int _{-\pi / 2}^{\pi / 2}[f(x)+f(-x)][g(x)-g(-x)] d x$

Let $\varphi(x)=[f(x)+f(-x)][g(x)-g(-x)]$

$\Rightarrow \quad \varphi(-x)=[f(-x)+f(x)][g(-x)-g(x)]$

$\Rightarrow \quad \varphi(-x)=-\varphi(x)$

$\Rightarrow \varphi(x)$ is an odd function.

$$ \therefore \quad \int _{-\pi / 2}^{\pi / 2} \varphi(x) d x=0 $$

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