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AP EAMCET201825 Apr 2018Morning ShiftMathematicsIndefinite IntegrationActual

∫ d x tan x + cot x + sec x + cosec x =

Options

  1. A1 2 ( sin x - cos x + x ) + c
  2. B1 2 ( sin x - cos x - x ) + c
  3. C1 2 ( sin x - cos x - tan x + cot x ) + c
  4. D1 2 ( sin x + cos x - tan x - cot x ) + c

Correct answer

B. 1 2 ( sin x - cos x - x ) + c

Step-by-step solution

Let, I = ∫ d x tan x + cot x + sec x + cosec x = ∫ d x sin x cos x + cos x sin x + 1 cos x + 1 sin x = ∫ sin x cos x sin 2 x + cos 2 x + sin x + cos x d x = ∫ sin x cos x 1 + sin x + cos x d x = ∫ sin x cos x 1 + sin x + cos x × sin x + cos x - 1 sin x + cos x - 1 d x = ∫ sin x cos x sin x + cos x - 1 sin x + cos x 2 - 1 d x = ∫ sin x cos x sin x + cos x - 1 sin 2 x + cos 2 x + 2 sin x cos x - 1 d x = ∫ sin x cos x sin x + cos x - 1 2 sin x cos x d x = 1 2 &#8747

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