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## Homework Statement

find

[tex]\int\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)}dx[/tex]

## Homework Equations

None

## The Attempt at a Solution

[tex]\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^{2}} + \frac{Cx + D}{x^{2}+1}[/tex]

[tex]x^{2}-2x-1 = A(x-1)(x^{2}+1) + B(x^{2}+1) + (Cx + D)(x-1)^{2}[/tex]

[tex]x^{2}-2x-1 = Ax^{3}-Ax^{2}+Ax-A+Bx^{2}+B+Cx^{3}-2Cx^{2}+Cx+Dx^{2}-2Dx+D[/tex]

[tex]x^{2}-2x-1 = x^{3}(A+B+C) + x^{2}(-A+B-2C+D) + x(A+C-2D) - A + B +D[/tex]

so A+B+C = 0, -A+B-2C+D = 1, A+C-2D=-2 and -A+B+D=-1

solving for coefficients, we get

A = 5/3

B = -2/3

C = -1

D = 4/3

so [tex]\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)} = \frac{5/3}{x-1} - \frac{2/3}{(x-1)^{2}} - \frac{x-(4/3)}{x^{2}+1}[/tex]

but apparently, the last statement is not correct. Did I break down the rational function incorrectly?

Thank you all in advance for your help!