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If ∫sin x1 t2...

Question

If $\int_{s i n x}^{1} t^{2} . f (t) d t = 1 - s i n x, \forall x ϵ (0, \frac{π}{2})$ then the value of $f (\frac{1}{\sqrt{3}})$ is

A

\frac{1}{\sqrt{3}}

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B

\sqrt{3}

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C

\frac{1}{3}

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D

3

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Solution

The correct option is D $3$
$\int_{s i n x}^{1} t^{2} . f (t) d t = 1 - s i n x$
Differentiating both sides with respect to ‘x’
$0 - s i n^{2} x . f (s i n x) . c o s x = - c o s x \Rightarrow c o s x [1 - s i n^{2} x . f (s i n x)] = 0 B u t c o s x \neq 0$
so, $f (s i n x) = \frac{1}{s i n^{2} x} f (\frac{1}{\sqrt{3}}) = 3$

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