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JEE Main2018MathematicsIndefinite IntegrationActual

If f x - 4 x + 2 = 2 x + 1 , x ∈ R - 1 , - 2 , then ∫ f x d x is equal to

Options

  1. A12 ln 1 - x - 3 x + C  
  2. B- 12 ln 1 - x - 3 x + C
  3. C12 ln 1 - x + 3 x + C
  4. D- 12 ln 1 - x + 3 x + C

Correct answer

B. - 12 ln 1 - x - 3 x + C

Step-by-step solution

We have, f x - 4 x + 2 = 2 x + 1 Let x - 4 x + 2 = X ⇒ x X + 2 X = x - 4 ⇒ x X - 1 = - 4 - 2 X ⇒ x = 2 2 + X 1 - X Putting x = 2 2 + X 1 - X in the given functional equation, we get ⇒ f X = 2 × 2 2 + X 1 - X + 1 = 8 + 4 X + 1 - X 1 - X = 3 X + 9 1 - X = 3 X + 3 1 - X Replacing X → x , we get f x = 3 x + 3 1 - x ⇒ ∫ f x d x = 3 ∫ x + 3 1 - x d x = 3 ∫ 4 - 1 - x 1 - x d x = 3 ∫ 4 1 - x d x - ∫ d x = - 12 ln 1 - x - 3 x + C , where C is the constan

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