Indefinite Integration 1 Question 37
38. 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
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Solution:
- 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.
$$ \therefore \quad \int _{-\pi / 2}^{\pi / 2} \varphi(x) d x=0 $$
