Evaluate : `int_(0)^(1) cot^(-1) (1-x + x^(2))dx`

Options

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

Answer

B

Explanation

`int_(0)^(1) co^(-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) - (In 2)/(2)) = (pi)/(2) - In 2`