Calculus: Early Transcendentals, Chapter 7, 7.4, Section 7.4, Problem 22

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