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`int (x^2 + 2x - 1)/(x^3 - x) dx` Evaluate the integral

Integrate `int(x^2+2x-1)/(x^3-x)dx`


Rewrite the rational function using partial fractions.


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


`x^2+2x-1=A(x^2-1)+Bx(x-1)+Cx(x+1)`


`x^2+2x-1=Ax^2-A+Bx^2-Bx+Cx^2+Cx`


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


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


`-A=-1` 


`A=1`



` ` `A+B+C=1`


`1+B+C=1`


`B+C=0`



`C-B=2`


`C+B=0`


`2C=2`


`C=1`



`B+C=0`


`B+1=0`


`B=-1`



`int(x^2+2x-1)/(x^3-x)dx=int(1/x)dx-int1/(x+1)dx+1/(x-1)dx`


`=ln|x|-ln|x+1|+ln|x-1|+C`


`=ln|[x(x-1)]/(x+1)|+C`



The final answer is:


`=ln|[x(x-1)]/(x+1)|+C `

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