Question

Evaluate: `int((x-1)^2)/(x^4+x^2+1) dx`

Answer 1

`I=int(x^(2)-2x+1)/(x^(4)+x^(2)+1)dx`
` =int(x^(2)+1)/(x^(4)+x^(2)+1)dx-int(2x dx)/(x^(4)+x^(2)+1)`
`=int((1+(1)/(x^(2)))dx)/(x^(2)+1+(1)/(x^(2)))-int(dt)/(t^(2)+t+1), " where " t=x^(2)`
`=int ((1+(1)/(x^(2)))dx)/((x-(1)/(x))^(2)+3)-int(dt)/((1+(1)/(2))^(2)+(3)/(4))`
`=(1)/(sqrt(3)) "tan"^(-1)(x-(1)/(x))/(sqrt(3))-(1)/((sqrt(3))/(2))"tan"^(-1)(t+(1)/(2))/((sqrt(3))/(2))+c`
`=(1)/(sqrt(3))"tan"^(-1)(x^(2)-1)/(sqrt(3)x)-(2)/(sqrt(3))"tan"^(-1)(2x^(2)+1)/(sqrt(3))+c`