Integration of Rational Functions

[forestmine]

Homework Statement

Evaluate the integral. (Remember to use ln |u| where appropriate.)

∫ds/s^2(s − 1)^2

Homework Equations

The Attempt at a Solution

I attempted a solution using the method of partial fractions, but it seems my answer is wrong. Here's what I did...

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

Then multiplying by a common denominator,

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

Equating the coefficients and solving for A, B, C, and D, I get A=1, B=1, C=-1, D=3

So my integral now looks like

∫1/s+1/s^2-1/(s-1)+3/(s-1)^2

And taking the integral, I got

lns - 1/s - ln(s-1) + 3/(s-1)

which evidently is wrong.

If anyone can point me in the right direction in terms of where I went wrong, it would be greatly appreciated.

Thanks!

 

[SammyS]

Homework Statement

Evaluate the integral. (Remember to use ln |u| where appropriate.)

∫ds/(s^2(s − 1)^2)

Homework Equations

The Attempt at a Solution

I attempted a solution using the method of partial fractions, but it seems my answer is wrong. Here's what I did...

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

Then multiplying by a common denominator,

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

Equating the coefficients and solving for A, B, C, and D, I get A=1, B=1, C=-1, D=3

So my integral now looks like

∫1/s+1/s^2-1/(s-1)+3/(s-1)^2

And taking the integral, I got

lns - 1/s - ln(s-1) + 3/(s-1)

which evidently is wrong.

If anyone can point me in the right direction in terms of where I went wrong, it would be greatly appreciated.

Thanks!

There is an error in your partial fraction decomposition. D is incorrect.

 

[forestmine]

Thanks for the reply!

So for s^2, I got -2A - 2B - C + D = 0

plugging in A, B, and C:

-2(1) - 2(1) - (-1) = 0

and I get D=3?

I'm not seeing where I'm going wrong...

 

[ehild]

Recalculate your parameters and show your work. (I got A=2, B=1, C=-2, D=1. )

ehild

 

[forestmine]

Wow, I wonder if I'm going about this all wrong.

Here's what I did:

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

For x^0 = 1=B

For x = 0 = A + B ----> A=-B=-1 (But that contradicts your answer)

For x^2 = 0 = -2A -2B - C + D

For x^3 = 0 = A + C ----> C=1

Looks like all of my values are wrong. What am I doing incorrectly?

 

[SammyS]

Wow, I wonder if I'm going about this all wrong.

Here's what I did:

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

Where does the above line come from.

Seems like a strange place to start --- like you're skipping some steps.

 

[forestmine]

It comes from here:

1/s^2(s − 1)^2=A/s +B/s^2+C/(s-1)+D/(s-1)^2

And then multiplying all of the above by s^2(s − 1)^2

 

[ehild]

Wow, I wonder if I'm going about this all wrong.

Here's what I did:

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

For x^0 = 1=B

For x = 0 = A + B ----> A=-B=-1 (But that contradicts your answer)

For s^1: A-2B=0 A=-2B=-2.


ehild

 

[forestmine]

Got it -- I was in fact equating the coefficients incorrectly.

Thanks so much for all the help!