SATHEE: Indefinite Integration 3 Question 5

Indefinite Integration 3 Question 5

5. If $\int_{\sin x}^{1} t^{2} f (t) d t = 1 - \sin x, \forall x \in (0, π / 2)$ , then $f \frac{1}{\sqrt{3}}$ is

(a) 3

(b) $\sqrt{3}$

(c) $1 / 3$

(d) None of these

(2005, 1M)

Show Answer

Answer:

Correct Answer: 5. (a)

Solution:

Since $\int_{\sin x}^{1} t^{2} f (t) d t = 1 - \sin x$ , thus to find $f (x)$ .

On differentiating both sides using Newton Leibnitz formula

i.e. $\frac{d}{d x} \int_{\sin x}^{1} t^{2} f (t) d t = \frac{d}{d x} (1 - \sin x)$

$\Rightarrow 1^{2} f (1) \cdot (0) - (\sin^{2} x) \cdot f (\sin x) \cdot \cos x = - \cos x$

$\Rightarrow f (\sin x) = \frac{1}{\sin^{2} x}$

For $f \frac{1}{\sqrt{3}}$ is obtained when $\sin x = 1 / \sqrt{3}$

$i.e. f \frac{1}{\sqrt{3}} = (\sqrt{3})^{2} = 3$

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