For real number `u, (-pi)/(2)lt tan^(-1)u lt (pi)/(2)` and `0 lt cot^(-1)u lt pi`, then the value of `(int_(0)^(1)cot^(-1)(l-x+x^(2))dx)/(int_(0)^(1)tan^(-1)xdx)`, is :

Options

A. log 2
B. `(pi)/(2)+log2`
C. log 4
D. `(pi)/(2)-log4`

Answer

A

Explanation

We have,
`2overset(1)underset(0)inttan^(-1)xdx=overset(1)underset(2)cot^(-1)(1-x+x^(2))dx " "`....(i)
`2overset(1)underset(0)int tan^(-1)xdx =overset(1)underset(0)int {(pi)/(2)-tan^(-1)(1-x+x^(2))}dx`
`overset(1)underset(0)inttan^(-1)(1-x+x^(2))dx=(pi)/(2)-2overset(1)underset(0)int tan^(-1)xdx`
`rArr overset(1)underset(0)int tan^(-1)(1-x+x^(2))dx=(pi)/(2)-2[x tan^(-1) x-(1)/(2)log(1+x^(2))]_(10)^(1)`
`rArr underset(0)overset(1)int tan^(-1)(1-x+x^(2))dx=(pi)/(2)-2((pi)/(4)-(1)/(2)log2)=log2`