Evaluate `int_(0)^(pi//4) log (1+tan theta)d theta`

Options

A. `(pi)/(8)log_(e )2`
B. `(pi)/(4)log_(e )2`
C. `-(pi)/(8)log_(e )2`
D. `-(pi)/(4) log_(e ) 2`

Answer

A

Explanation

We have,
`I=underset(0)overset(pi//4)int log(1+tanx)dx" "`…..(i)
`rArr I=underset(0)overset(pi//4)int log{1+tan((pi)/(4)-x)}" "`[Using Prop.IV]
`rArr I=underset(0)overset(pi//4)int log(1+(1-tanx)/(1+tanx))dx`
`rArr I=underset(0)overset(pi//4)int log((2)/(1+tanx))dx" "`....(ii)
Adding (i) and (ii), we get
`2I=underset(0)overset(pi//4)int log 2 dx=(pi)/(4)log_(e )2rArrI=(pi)/(8)log_(e )2`