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

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

Text Solution

Verified by Experts

`I=int(x^(2)-1)/((x^(4)+3x^(2)+1)tan^(-1)(x+(1)/(x)))dx`
`=int(1-(1)/(x^(2)))/((x^(2)+(1)/(x^(2))+3)tan^(-1)(x+(1)/(x)))dx`
Put `x+(1)/(x)=t`
` :. (1-(1)/(x^(2)))dx=dt " and " x^(2)+(1)/(x^(2))+3=t^(2)+1`
` :. I=int(dt)/((t^(2)+1)tan^(-1)t)="In"|tan^(-1)t|+C`
`="In"|tan^(-1)(x+(1)/(x))|+C`
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int(x^2-1)/(x^4+3x^2+1)dx(x>0) is

int ((x^2 -1)dx)/((x^4 + 3x^2 + 1) tan^-1((x^2 + 1)/x)) = k logabs(tan^-1((x^2 +1)/x)) + c