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Integration Using Partial Fractions

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Integrate (x^3 - 8x^2 - 1)/((x+3)(x^2-4x+5))

    2. Relevant equations

    This is an integration by partial fractions.

    3. The attempt at a solution

    http://www.wolframalpha.com/input/?i=integral+%28%28x^3-8x^2-1%29%2F%28%28x%2B3%29%28x^2-4x%2B5%29%29%29dx

    I understand everything except where the integrand is rewritten after finding A, B, and C of the partial fraction decomposition. If anyone can help me understand how the integrand is rewritten that would be great. I just cannot make any sense out of it.

    Thanks

    Edit: If anyone is unfamiliar with WolframAlpha there is a "show steps" button in the top right corner of the problem statement, this is what I am referring to. It is about the 4th step down, rewrite the integrand.
     
    Last edited: Oct 3, 2011
  2. jcsd
  3. Oct 3, 2011 #2
    Here is a screenshot of the part I am asking about.
     

    Attached Files:

  4. Oct 3, 2011 #3

    Mark44

    Staff: Mentor

    They're starting with this part of the problem:
    [tex]\int \frac{14 - 41x}{x^2 - 4x + 5}dx[/tex]

    What they're doing is manipulating things that that a substitution of u = x2 - 4x + 5 (hence du = (2x - 4) dx) will work.

    To get a numerator of -41/2*x + 82, which equals -(41/2)(2x - 4), they need to keep the numerator unchanged, so they are subtracting 68.

    14 - 41x = -41x + 82 - 68 = -41/2(2x - 4) - 68
     
  5. Oct 3, 2011 #4
    Yep I see it. I noticed the differential so I thought it was a substitution but I just could not get my head around the manipulation. Thank you for breaking it down for me.
     
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