The value of the integral `int_(0)^(1)xcot^(-1)(1-x^(2)+x^(4))dx` is

Options

A. `(pi)/(4)-(1)/(2)log_(e )2`
B. `(pi)/(2)-log_(e )2`
C. `(pi)/(2)-(1)/(2)log_(e )2`
D. `(pi)/(4)-log_(e )2`

Answer

A

Explanation

`I=int_(0)^(1)cot^(-1)(1-x^(2)+x^(4))dx`
`I=int_(0)^(1)x tan^(-1)((1)/(1-x^(2)+x^(4)))dx`
Let
`x^(2)=t`
`2xdx = dt`
`=int_(0)^(1)tan^(-1)((1)/(1-t+t^(2))).(dt)/(2)`
`=(-1)/(2)int_(0)^(1)tan^(-1)(((t-1)-t)/(1-t(t-1)))dt`
`I=-(1)/(2)int_(0)^(1)(tan^(-1)(t-1)-tan^(-1)t)dt " "` ........(1)
By prop `4 int_(0)^(1)f(x)dn=int_(0)^(a)f(a-x)dx`
`I=-(1)/(2)int_(0)^(1)(-tan^(-1)t-tan^(-1)(t-1))dt " "` ....... (2)
Adding `1 + 2 rArr 2I = int_(0)^(1)(tan^(-1)t-tan^(-1)(t-1))dt`
`2I=int_(0)^(1)tan^(-1)tdt + int_(0)^(1)tan^(-1)dt`
`therefore I = int_(0)^(1)1_(II).tan_(I)^(-1)+dt`
`I=(tan^(-1)t.t)_(0)^(1)-int_(0)^(1)(1)/(1+t^(2)).tdt`
`I=(pi)/(4)-(1)/(2)[ln(1+t^(2))]_(0)^(1)`
`I=(pi)/(4)-(1)/(2)ln2`