The value of the integral ` underset(0)overset(1)int cot^(-1) (1-x+x^(2))dx`, is

Options

A. `(pi)/2+In2`
B. `(pi)/2-In2`
C. `pi-In2`
D. `pi+In2`

Answer

B

Explanation

`int_(0)^(1) cot^(-1) (1- x + x^(2)) dx = int_(0)^(1) tan^(-1) .(1)/(1-x(1-x))dx`
`=int_(0)^(1) {tan^(-1) x + tan^(-1)(1-x)} dx = 2 int_(0)^(1) tan^(-1) x dx`
` = 2 { x tan^(-1) x - (1)/(2) In (x^(2) + 1)}_(0)^(1) = 2 ((pi)/(4) - (In2)/(2)) = (pi)/(2) - In 2`