SATHEE: Indefinite Integration 1 Question 8

Indefinite Integration 1 Question 8

9. Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f (x) = f (a - x)$ and $g (x) + g (a - x) = 4$ , then $\int_{0}^{a} f (x) g (x) d x$ is equal to

(2019 Main, 12 Jan I)

(a) $4 \int_{0}^{a} f (x) d x$

(b) $\int_{0}^{a} f (x) d x$

(d) $- 3 \int_{0}^{a} f (x) d x$

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Solution:

Let $I = \int_{0}^{a} f (x) g (x) d x$

$= \int_{0}^{a} f (a - x) g (a - x) d x$

$∵ \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$

$\Rightarrow I = \int_{0}^{a} f (x) [4 - g (x)] d x$

$[∵ f (x) = f (a - x) and g (x) + g (a - x) = 4]$

$= \int_{0}^{a} 4 f (x) d x - \int_{0}^{a} f (x) g (x) d x$

$\Rightarrow I = 4 \int_{0}^{a} f (x) d x - I$

[from Eq. (i)]

$\Rightarrow 2 I = 4 \int_{0}^{a} f (x) d x \Rightarrow I = 2 \int_{0}^{a} f (x) d x$ .

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Indefinite Integration 1 Question 8

9. Let f and g be continuous functions on [0,a] such that f(x)=f(a−x) and g(x)+g(a−x)=4, then ∫0af(x)g(x)dx is equal to

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9. Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f (x) = f (a - x)$ and $g (x) + g (a - x) = 4$ , then $\int_{0}^{a} f (x) g (x) d x$ is equal to