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JEE Advanced2015MathematicsDefinite IntegrationActual

If α = ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x 12 + 9 x 2 1 + x 2 d x where tan - 1 ⁡ x takes only principal values, then the value of log e ⁡ 1 + α - 3 π 4 is

Correct answer

0

Step-by-step solution

α =   ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x   12 + 9 x 2 1 + x 2   d x =   ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x   9 + 3 1 + x 2   d x =   e 9 x + 3 tan - 1 ⁡ x | 0 1 =   e 9 + 3 π 4 - 1 ⇒    ln   ( 1 + α ) -   3 π 4 = ln 1 + e 9 + 3 π 4 - 1 - 3 π 4 = ln e 9 + 3 π 4 - 3 π 4 = 9 .

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