Let f be a differentiable function in (0, /2). If _ x ^ 1 t^ 2 f(t) dt = ^ 3 x + cos x, then (1/ 3 )f'(1/ 3 ) is equal to

JEE Main 2022 Mathematics question

Let f be a differentiable function in (0, ). If x}^{1} dt = + cos x, then is equal to

(1)
(2)Correct
(3)
(4)

Answer: (2)

Solution

Differentiating both sides with respect to x: the left side gives f(cos x)(−sin x) = sin x f(cos x), and the right side gives cos x − sin x. So f(cos x) = 3 tan . Differentiating again, −sin x \cos\sec ^{2}$x -

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Question ID VK-034513