Partial Fractions Help - Calc II Integration

[demersal]

Homework Statement

\int1/(s^{2}(s-1)^{2}) ds

Homework Equations

Partial Fractions

The Attempt at a Solution

= \frac{A}{s^{2}}+\frac{B}{s-1}+\frac{C}{(s-1)^{2}}

Setting numerators equal to each other:
1 = A(s-1)(s-1)^{2} + Bs^{2}(s-1)^{2}+Cs^{2}(s-1)
1=(As-A)(s^{2}-2s+1)+Bs^{2}(s^{2}-2s+1)+Cs^{3}+Cs^{2}
1=As^{3}-2As^{2}+As-As^{2}+2As-A+Bs^{4}-2Bs^{3}+Bs^{2}+Cs^{3}-Cs^{3}+Cs^{2}
1=Bs^{4}+(A-2B+C)s^{3}+(-3A+B-C)s^{2}+3As-A

Using coefficients to solve for variables (here's where something is up!)
s^{4}: 0=B
s^{3}: 0=A-2B+C
s^{2}: 0=-3A+B-C
s: 0=3A
#: 1=-A

They don't add up! Am I doing something wrong?
Thank you for everyone's time and help!

 

[Dick]

Don't you need a D/s term as well? If you can't solve it you are missing a variable. (s+1)^2 is a repeated factor and you have two variables for it. Why not for s^2 as well?

 

[demersal]

Ohh okay! I'll give it a try! :)

EDIT: Adding a D/s term only seemed to make the situation worse. Unless I made a slight error, there is now no term A, B, C, or D that stands unattached to an "s" and thus there is no variable to equate to the constant.

Ideas??

 

[bigevil]

I'm not sure if you really have to do it by partial fractions, but a simpler way to do this problem is by trigonometric substitution.

 

[HallsofIvy]

\frac{1}{s^2(s-1)^2}= \frac{A}{s}+ \frac{B}{s^2}+ \frac{C}{s-1}+ \frac{D}{(s-1)^2}
1= As(s-1)^2+ B(s-1)^2+ Cs^2(s-1)+ Ds^2
Taking s= 0, B= 1.
Taking s= 1, D= 1.
Taking s= -1, 1= -4A+ 4B- 2C+ D= -4A+ 4- 2C+ 1 so 4A+ 2C= 4.
Taking s= 2, 1= 2A+ B+ 4C+ 4D= 2A+ 1+ 4C+ 4 so 2A+ 4C= -3

 

[demersal]

Thank you so much, Hall of Ivy! I had a feeling I was doing way too much work. Now I understand :)