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Integrate: (x^2 + 4)/ (x^2+5x-6) dx

  1. Nov 10, 2012 #1
    1. The problem statement, all variables and given/known data

    "Express the integrand (what does "integrand" mean?) as a sum of partial fractions and evaluate the integrals.

    ∫(x + 4)/ (x^2+5x-6) dx

    2. Relevant equations



    3. The attempt at a solution

    x^2+5x-6 = (x-1)(x+6)

    Gives:

    ∫ A/(x-1) + B/(x+6) dx

    Findig A and B:

    A(x+6) + B(x-1)

    A+B=1

    6A-B =4

    A= 5/7
    B= - (2/7)


    Then:

    ∫ (5/7) (1/x-1) dx +∫ - (3/7)(1/x+6) dx

    Gives I think:

    5/7ln(x-1) -2/7ln(x+6)


    But its wrong accordng to the solution. I can post the solution here if you want me to:
     
    Last edited: Nov 10, 2012
  2. jcsd
  3. Nov 10, 2012 #2

    Mentallic

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    The integrand is the expression being integrated, in this case, the rational function [tex]\frac{x+4}{x^2+5x-6}[/tex]

    How did you get A+B=1 ?
     
  4. Nov 10, 2012 #3
    Okey, thanks.

    A(x+6) + B(x-1) = x+4

    Ax +Bx +6A -B = x+4

    Ax + Bx = x


    A+B =1
     
  5. Nov 10, 2012 #4

    Mentallic

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    Oh I see, you were comparing coefficients :smile:

    Sorry I didn't spot your error before, but you incorrectly solved the simultaneous equations in A and B. 1 - 5/7 = 2/7
     
  6. Nov 10, 2012 #5
    Oh, I changed it now. It is correct so far, but I dont understand how they get


    5/7ln(x-1) -2/7ln(x+6)


    to become ( from solution):


    (1/7)ln[(x+6)^2(x-1)^5] + C
     
  7. Nov 10, 2012 #6

    SammyS

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    That should be
    5/7ln(x-1) + 2/7ln(x+6)​
    Then use properties of logarithms to get the desired result.

    Of course the given answer includes the constant of integration.
     
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