Question

The integral `int _(0)^(1) (dx)/(1-x+x^(2)) ` has the value
A. `(2pi)/3`
B. `(2pi)/(sqrt(3))`
C. `(2pi)/(3sqrt(3))`
D. None of these

Answer 1

Correct Answer - C
`int_(0)^(1)(dx)/(x^(2)-x+1)=int_(0)^(1)(dx)/((x-(1)/(2))^(2)+((sqrt(3))/(2))^(2))`
`=(2)/(sqrt(3))[tan^(-1)((2x-1)/(sqrt(3)))]_(0)^(1)`
`= (2)/(sqrt(3))[tan^(-1)((1)/(sqrt(3)))-tan^(-1)(-(1)/(sqrt(3)))]`
`=(2)/(sqrt(3))[(pi)/(6)-(-(pi)/(6))]=(2)/(sqrt(3)).(pi)/(3)=(2pi)/(3sqrt(3))`