Evaluate : `int_0^(pi/4)log(1+tanx)dxdot`

Options

A. `(pi)/(4)`
B. `(pi)/(4)log2`
C. `(pi)/(8)log2`
D. `0`

Explanation

`I=int_(0)^(pi//4)log(1+tanx)dx`……..`(i)`
`I=int_(0)^(pi//4)log[1+tan((pi)/(4)-x)]dx=int_(0)^(pi//4)log{1+(1-tanx)/(1+tanx)}dx`
`=int_(0)^(pi//4){log2-log(1+ranx)}dx=int_(0)^(pi//4)(log2)dx-I`
`implies2I=(log2)[x]_(0)^(pi//4)=(pi)/(4)(log2)impliesI=(pi)/(8)(log2)`.