1. Limited time only! Sign up for a free 30min personal 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 by Partial Fractions

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    ∫(x3+4)/(x2+4)dx

    2. Relevant equations

    n/a

    3. The attempt at a solution

    I know I have to do long division before I can break this one up into partial fractions. So I x3+4 by x2+4 and got x with a remainder of -4x+4 to be written as x+(4-4x/x2+4).

    Then I rewrote the initial integral as ∫x + ∫(4-4x)/(x2+4) but then I get stuck from there.
    Should the partial fraction be ∫x + ∫(Ax+B)/(x2+4)?
    And if so, how would I go about solving that?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 20, 2012 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    ##\frac x {x^2+4}## can be done with a u-substitution and ##\frac 2 {x^2+4}##is an Arctan form.
     
  4. Mar 20, 2012 #3
    How/where are you getting x/x2+4 and 2/x2+4 from?
     
  5. Mar 20, 2012 #4
    Long division for
    [itex]\frac{x^3+4}{x^2+4}[/itex]

    yields
    [itex]x - 4(\frac{x-1}{x^2+4})[/itex].

    As you mentioned, you now have
    [itex]\int x dx - 4\int \frac{x-1}{x^2+4} dx[/itex].

    For the second integral, split up the integrand:
    [itex]\int x dx - 4(\int \frac{x}{x^2+4} dx - \int \frac{1}{x^2+4} dx)[/itex].

    (This is what LCKurtz did, at least.)
     
    Last edited: Mar 20, 2012
  6. Mar 21, 2012 #5
    So do you think it was just a typo that he wrote 2/x2+4 instead of 1/x2+4 because that 2 really confused me
     
  7. Mar 21, 2012 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, the 2 was a typo. At any rate a constant factor wouldn't affect the method of integration.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook