SATHEE: Definite Integration Question 37

Definite Integration Question 37

Question 37

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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Definite Integration Question 37

Question 37

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