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AP EAMCET202119 Aug 2021Evening ShiftMathematicsIndefinite IntegrationActual

∫ ( cos x ) logcot x 2 d x =

Options

  1. A( sin ⁡ x ) log ⁡ cot ⁡ x 2 + c
  2. B( cos ⁡ x ) log ⁡ cot ⁡ x 2 + c
  3. C( sin ⁡ x ) log ⁡ cot ⁡ x 2 + x + c
  4. D( sin ⁡ x ) log ⁡ cot ⁡ x 2 − x + c

Correct answer

C. ( sin ⁡ x ) log ⁡ cot ⁡ x 2 + x + c

Step-by-step solution

Applying integration by parts ∫ cos x · log   cot x 2 d x = log   cot x 2 ∫ cos x d x - ∫ d log   cot x 2 d x × ∫ cos x   d x d x = log   cot x 2 ∫ cos x d x - ∫ 1 c o t x 2 × d d x c o t x 2 × ∫ cos x   d x d x = sin x · log   cot x 2 - ∫ 1 c o t x 2 × - cos e c 2 x 2 × 1 2 × ∫ cos x   d x d x = sin x · log   cot x 2 + ∫ 1 2 · sin   x 2 cos   x 2 &#215

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