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Evaluate int (x^(2)-sqrt(3)x+1)/(x^(4)-...

Evaluate `int (x^(2)-sqrt(3)x+1)/(x^(4)-x^(2)+1)dx`

Text Solution

Verified by Experts

The correct Answer is:
`2 tan^(-1)(2x+sqrt(3))+C`
`int (x^(2)-sqrt(3x)+1)/(x^(4)-x^(2)+1)dx=int (x^(2)-sqrt(3x)+1)/((x^(2)-1)^(2)-3x^(2))dx`
`=int (x^(2)-sqrt(3x)+1)/((x^(2)-sqrt(3)x+1)(x^(2)+sqrt(3)x+1))dx`
`=int(1)/(x^(2)+sqrt(3)x+1)dx`
`=int (1)/((x+(sqrt(3))/(2))^(2)+(1)/(4))dx`
`=(1)/(1//2)"tan"^(-1)(x+(sqrt(3))/(2))/((1)/(2))+C`
`=2 tan^(-1)(2x+sqrt(3))+C`
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