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

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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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