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 ` **

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now