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JEE Main2018MathematicsIndefinite IntegrationActual

If ∫ tan x 1 + tan x + tan 2 x d x = x - K A tan - 1 K tan x + 1 A + C , ( C is a constant of integration), then the ordered pair K , A is equal to

Options

  1. A2 , 1
  2. B2 , 3
  3. C- 2 , 1
  4. D- 2 , 3

Correct answer

B. 2 , 3

Step-by-step solution

Let I = ∫ tan x 1 + tan x + tan 2 x d x ⇒ I = ∫ 1 - sec 2 x 1 + tan x + tan 2 x d x ⇒ I = x - ∫ d t 1 + t + t 2 Where, tan x = t ⇒ sec 2 x d x = d t ∴   I = x - 1 3 2 tan - 1 t + 1 2 3 2 + C , where C is the constant of integration. = x - 2 3 tan - 1 2 tan x + 1 3 + C ∴   K = 2   &   A = 3 .

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