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JEE Main202116 Mar 2021Morning ShiftMathematicsDefinite IntegrationActual

Let f : 0 , 2 → R be defined as f x = log 2 1 + tan π x 4 . Then, lim n → ∞ 2 n f 1 n + f 2 n + … . + f 1 is equal to ________.

Correct answer

0

Step-by-step solution

Given f x = log 2 1 + tan π x 4 = log e 1 + tan π x 4 log e 2 E = 2 lim n → ∞ ∑ r = 1 n 1 n f r n E = 2 log e 2 ∫ 0 1 log e 1 + tan π x 4 d x . . . ( i ) Replacing x → 1 - x E = 2 log e 2 ∫ 0 1 log e 1 + tan π 4 1 - x d x E = 2 log e 2 ∫ 0 1 log e 1 + tan π 4 - π 4 x d x E = 2 log e 2 ∫ 0 1 log e 1 + 1 - tan π 4 x 1 + tan π 4 x d x E = 2 log e 2 ∫ 0 1 log e 2 1 + tan π x 4 d x E = 2 log e 2 ∫ 0 1 log e 2 - log e

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