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`int (x^2 - 2x - 1)/((x - 1)^2(x^2 + 1)) dx` Evaluate the integral

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Integrate `int(x^2-2x-1)/[(x-1)^2(x^2+1)]dx`

Rewrite the rational function using partial fractions.





`x^2-2x-1=Ax^3-Ax^2-Ax-A+Bx^2+B `

`+ Cx^3-2Cx^2+Cx+Dx^2-2Dx+D `


`x^2-2x-1=(A+C)x^3+(-A+B-2C+D)x^2 `



Equate coefficients and solve for A, B, C, and D.











`B=C `
















The first integral follows the pattern `int(du)/u=ln|u|+C`

` `



Integrate the second integral using u-substitution.

Let `u=x-1`








Integrate the third integral using u-subsitution.

Let `u=x^2+1`

` `








The fourth integral matches the pattern





The final answer is:






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