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BITSAT2018MathematicsDefinite IntegrationActual

The value of ∫ 0 1 cot - 1 1 - x + x 2 d x is

Options

  1. Alog e 2
  2. Bπ 2 - log e 2
  3. Cπ 2 + log e 2
  4. D- log e 2

Correct answer

B. π 2 - log e 2

Step-by-step solution

cot - 1 1 - x + x 2 = tan - 1 1 1 - x + x 2 ⇒ cot - 1 1 - x + x 2 = tan - 1 1 1 - x 1 - x ⇒ cot - 1 1 - x + x 2 = tan - 1 x + 1 - x 1 - x 1 - x ⇒ cot - 1 1 - x + x 2 = tan - 1 x + tan - 1 1 - x ∴   ∫ 0 1 cot - 1 1 - x + x 2 d x = ∫ 0 1 tan - 1 x d x + ∫ 0 1 tan - 1 1 - x d x ⇒ ∫ 0 1 cot - 1 1 - x + x 2 d x = ∫ 0 1 tan - 1 x d x + ∫ 0 1 tan - 1 x d x ∵   ∫ 0 a f x d x = ∫ 0 a f a - x d x ⇒ ∫ 0 1 cot - 1 1 - x +

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