Ex 7.11, 8 - Evaluate integral log (1 + tan x) dx

Ex 7.11,8 By using the properties of definite integrals, evaluate the integrals : 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯𝑑𝑥 Let I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯𝑑𝑥 ∴ I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮ 𝜋﷮4﷯−𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮ ﷯ 𝜋﷮4﷯ − tan﷮𝑥﷯﷮1 + tan﷮ ﷯ 𝜋﷮4﷯ . tan﷮𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ 1 − tan﷮𝑥﷯﷮1 + 1 . tan﷮𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1 − tan﷮𝑥﷯ + 1 − tan﷮𝑥﷯﷮1 + tan﷮𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 2﷮1 + tan﷮𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 2﷯﷯ − log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯﷯𝑑𝑥 I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 2﷯﷯﷯𝑑𝑥− 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I+I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯𝑑𝑥+ 0﷮ 𝜋﷮4﷯﷮ log﷮ 2﷯﷯﷯𝑑𝑥− 0﷮ 𝜋﷮4﷯﷮ log﷮ 1+ tan﷮𝑥﷯﷯﷯﷯ 2I= 0﷮ 𝜋﷮4﷯﷮ log﷮ 2﷯﷯﷯𝑑𝑥 2I= log﷮ 2﷯﷯ 0﷮ 𝜋﷮4﷯﷮𝑑𝑥﷯ I= log﷮ 2﷯﷮2﷯ 𝑥﷯﷮0﷮ 𝜋﷮4﷯﷯ I= log﷮2﷯﷮2﷯ 𝜋﷮4﷯ − 0﷯ I= log﷮2﷯﷮2﷯× 𝜋﷮4﷯ 𝑰= 𝝅﷮𝟖﷯ 𝒍𝒐𝒈﷮𝟐﷯