_ x 2 ( _ x^3 ^ ( /2)^3 ( (2t^ 1/3 ) + (t^ 1/3 ))dt (x - 2 )^2 ) is equal to :

JEE Main 2024 Mathematics question

is equal to :

(A)9π²/8Correct
(B)11π²/10
(C)3π²/2
(D)5π²/9

Answer: (A)

Solution

The limit has the form 0/0. By L'Hôpital's rule (with Leibniz's rule for the integral) it equals lim −3x²[sin 2x + cos x]/(2(x − π/2)), which is again 0/0 at x = π/2 (sin π = 0

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Question ID VK-016455