`int_(1)^(e){((logx-1))/(1+(logx)^(2))}^(2)`dx is equal to

Options

A. `e/2`
B. `1/2`
C. `(e-2)/2`
D. None of these

Answer

3

Explanation

`I=underset(1)overset(e)int[((logx-1))/(1+(log)^(2))]^(2)dx`
put log x=t`rArrx=e^(t)`
`dx=e^(1)dt`
`I=underset(0)overset(1)inte^(1){(t-1)/(1-t^(2))}^(2)dt`
`=underset(0)overset(1)inte^(t)[1/(1+t^(2))-(2t)/(1+t^(2))^(2)}dt`
`=[(e^(1))/(1+t^(2))]_(0)^(1)=e/2-1=(e-2)/2`