1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integration of Rational Functions

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data

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

    ∫ds/s^2(s − 1)^2


    2. Relevant equations



    3. 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!
     
  2. jcsd
  3. Feb 14, 2012 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    There is an error in your partial fraction decomposition. D is incorrect.
     
  4. Feb 14, 2012 #3
    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...
     
  5. Feb 14, 2012 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

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

    ehild
     
  6. Feb 14, 2012 #5
    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?
     
  7. Feb 14, 2012 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Where does the above line come from.

    Seems like a strange place to start --- like you're skipping some steps.
     
  8. Feb 14, 2012 #7
    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
     
  9. Feb 15, 2012 #8

    ehild

    User Avatar
    Homework Helper
    Gold Member

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


    ehild
     
  10. Feb 15, 2012 #9
    Got it -- I was in fact equating the coefficients incorrectly.

    Thanks so much for all the help!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integration of Rational Functions
Loading...