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

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

Text Solution

Verified by Experts

Putting `x^(2)=t" and "2xdx=dt` we get
`int(x)/((x^(4)-x^(2)+1))dx=(1)/(2)*int(dt)/((t^(2)-t+1))=(1)/(2)*int(dt)/({(t-(1)/(2))^(2)+((sqrt(3))/(2))^(2)})`
`=(1)/(2)*(1)/(((sqrt(3))/(2)))" tan "^(-1)((t-(1)/(2)))/(((sqrt(3))/(2)))+C`
`=(1)/(sqrt(3))*tan^(-1)((2t-1)/(sqrt(3)))+C=(1)/(sqrt(3))tan^(-1)((2x^(2)-1)/(sqrt(3)))+C`.
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