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JEE Main2017MathematicsIndefinite IntegrationActual

The integral ∫ 1 + 2 cot x cosec x + cot x d x , 0 < x < π 2 is equal to

Options

  1. A2 log sin x 2 + c
  2. B4 log sin x 2 + c
  3. C4 log cos x 2 + c
  4. D2 log cos x 2 + c

Correct answer

A. 2 log sin x 2 + c

Step-by-step solution

Let I = ∫ 1 + 2 cot x cosec   x + cot x d x     ⇒ I = ∫ 2 cot x cosec x + cosec 2 x + cot 2 x d x ⇒ I = ∫ cosec   x + cot   x 2 d x ⇒ I = ∫ cosec   x + cot   x d x ⇒ I = ∫ cosec   x + cot x d x ⇒ I = log tan x 2 + log sin x + c , where c is the constant of integration. ⇒ I = log sin x 2 cos x 2 · 2 sin x 2 cos x 2 + c ⇒ I = 2 log sin x 2 + c

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