`(i) int_(0)^(4)(dx)/(sqrt(x^(2)+2x+3))=int_(0)^(4)(dx)/(sqrt((x+1)^(2)+(sqrt(2))^(2)))`
`=[log|(x+1)+sqrt(x^(2)+2x+3)|]_(0)^(4)`
`={log|5+sqrt(27)|-log|1+sqrt(3)|}`.
`(ii) (dx)/((1+x+x^(2)))=int_(0)^(1)(dx)/([(x^(2)+x+(1)/(4))+(3)/(4)])=int_(0)^(1)(dx)/([(x+(1)/(2))^(2)+((sqrt(3))/(2))^(2)])`
`=[(2)/(sqrt(3))tan^(-1).((x+(1)/(2)))/((sqrt(3)//2))]_(0)^(1)=(2)/(sqrt(3))[tan^(-1)((2x+1)/(sqrt(3)))]_(0)^(1)`
`=(2)/(sqrt(3))[tan^(-1)(sqrt(3))-tan^(-1)((1)/(sqrt(3)))]`
`=(2)/(sqrt(3))*[(pi)/(3)-(pi)/(6)]=(pi)/(3sqrt(3))`.
