SATHEE: Indefinite Integration 1 Question 37

Indefinite Integration 1 Question 37

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

(1990, 2M)

(a) $π$

(b) 1

(c) -1

(d) 0

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Solution:

Let $I = \int_{- π / 2}^{π / 2} [f (x) + f (- x)] [g (x) - g (- x)] d x$

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

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

$\Rightarrow φ (- x) = - φ (x)$

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

$∴ \int_{- π / 2}^{π / 2} φ (x) d x = 0$

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Indefinite Integration 1 Question 37

38. Let f:R→R and g:R→R be continuous functions. Then, the value of the integral ∫−π/2π/2[f(x)+f(−x)][g(x)−g(−x)]dx is

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