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AP EAMCET202120 Aug 2021Evening ShiftMathematicsIndefinite IntegrationActual

If f ( x ) = 1 cos 2 x 1 + tan x , then its anti-derivative F ( x ) = , given F ( 0 ) = 4

Options

  1. A1 + tan x + 4
  2. B2 3 ( 1 + tan x ) 3 / 2
  3. C2 ( 1 + tan x + 1 )
  4. D1 + tan x + 2

Correct answer

C. 2 ( 1 + tan x + 1 )

Step-by-step solution

⇒ f x = 1 cos 2 x 1 + tan x p u t   1 + tan x = t ⇒ 1 · sec 2 x 2 · 1 + tan x = d t d x ⇒ d x = 2 1 + tan x sec 2 x d t ⇒ ∫ f x d x = ∫ 1 cos 2 x 1 + tan x d x ⇒ ∫ f x d x = ∫ 2 1 + tan x sec 2 x · cos 2 x · 1 + tan x d t ⇒ F x = ∫ 2 d t ⇒ F x = 2 t + C ⇒ F x = 2 1 + tan x + C ⇒ F 0 = 4 = 2 1 + tan 0 + C ⇒ C = 2 ⇒ F x = 2 1 + tan x + 2 ⇒ F x = 2 1 + tan x + 1

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