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Homework Help: Tricky integral using Partial Fractions

  1. Apr 11, 2010 #1
    1. The problem statement, all variables and given/known data
    The question asks me to express the integrand in partial fractions to evaluate the integral

    [tex] \int \frac{13x-4}{6x^{2} -x -2} dx [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Well 6x² -x - 2 doesn't factorise (or I cant see it factorised).

    So I tried doing long polynomial division and I got something that looked nothing like what the answer is. How should I begin to tackle this problem?

    Thanks
    Thomas
     
  2. jcsd
  3. Apr 11, 2010 #2
  4. Apr 11, 2010 #3
    6x² -x - 2 = 1/6 (6x + 3)(6x - 4)
     
  5. Apr 11, 2010 #4
    And as Gregg suggested you can remove the term linear in x (i.e. 13 x) term from the numerator before you start. That will save you work on the partial fraction expansion:

    You can write: 13 = 12 * (13/12)

    The numerator can thus be written as:

    13/12 (12 x - 48/13) =

    13/12 (12 x - 1 + 1 - 48/13) =

    13/12 (derivative of denominator -35/13) =

    13/12 derivative of denominator - 35/12

    So, the integral will be 13/12 Log[6 x^2 - x -2] plus the integral of

    -35/12 1/(6 x^2 - x -2)
     
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