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

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

Text Solution

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`I=int(x^(2)-1)/(x^(4)+x^(2)+1)dx`
`=int(1-(1)/(x^(2)))/(x^(2)+1+(1)/(x^(2)))dx`
`=int(1-(1)/(x^(2)))/((x+(1)/(x))^(2)-1^(2))dx`
Let `x+(1)/(x)=u. " Then " d(x+(1)/(x))=du " or "(1-(1)/(x^(2)))dx=du`
`" or " I=int(du)/(u^(2)-1^(2))`
`=(1)/(2(1))log|(u-1)/(u+1)|+C`
`=(1)/(2)log|(x+(1)/(x)-1)/(x+(1)/(x)+1)|+C=(1)/(2)log|(x^(2)-x+1)/(x^(2)+x+1)|+C`
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