Integrate `int(ds)/[s^2(s-1)^2]`

Integrate the given rational function using the method of partial fractions.

`1/[s^2(s-1)^2]=[A/s]+[B/s^2]+[C/(s-1)]+[D/(s-1)^2]`

` ` `1=As(s-1)^2+B(s-1)^2+Cs^2(s-1)+Ds^2`

`1=As(s^2-2s+1)+B(s^2-2s+1)+Cs^3-Cs^2+Ds^2`

`1=As^3-2As^2+As+Bs^2-2Bs+B+Cs^3-Cs^2+Ds^2`

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

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

`B=1`

`A-2B=0`

`A-2(1)=0`

`A=2`

`A+C=0`

`2+C=0`

`C=-2`

`-2A+B-C+D=0`

`-2(2)+1-(-2)+D=0`

`-4+1+2+D=0`

`D=1`

`int[2/s]+[1/s^2]+[-2/(s-1)]+[1/(s-1)^2]`

`=2ln|s|-(1/s)-2ln|s-1|-1/(s-1)+C`

**The final answer is: `=2ln|s|-(1/s)-2ln|s-1|-1/(s-1)+C ` **

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