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Difficult Integration Problem

  1. Sep 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Let [tex]f(x) = ax^2 + bx + c[/tex]
    [tex]a,b,c[/tex] are real numbers
    [tex]f(0) = 1[/tex] and [tex]\int\frac{f(x)}{x^2(x+1)^3}dx[/tex] is a rational function.

    Find [tex]f'(0)[/tex]

    2. Relevant equations



    3. The attempt at a solution

    [tex]f(0) = 1 = a(0)^2 + b(0) + c[/tex]
    [tex]c = 1[/tex]
    [tex]f'(x) = 2ax + b[/tex]
    [tex]f'(0) = 2a(0) + b = b[/tex]
    I tried to integrate [tex]\int\frac{f(x)}{x^2(x+1)^3}dx[/tex] and ended up doing some tedious partial fractions and ending up with logarithms (not rational?), so that's probably not the correct approach. I'm totally stuck at this point and any help would be greatly appreciated; Thanks!
     
  2. jcsd
  3. Sep 29, 2009 #2

    Mark44

    Staff: Mentor

    Your integrand looks like
    [tex]\frac{ax^2 + bx + 1}{x^2(x + 1)^3}~=~\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1} + \frac{D}{(x + 1)^2} + \frac{E}{(x + 1)^3} [/tex]

    After multiplying both sides by x2(x + 1)3 and grouping the terms by powers of x, I got these values for the constants A through E.
    A = b - 3
    B = 1
    C = -b + 3
    D = b - 4
    E = a - 3b + 7
    (Caveat: I didn't double-check my work, so there might be an error or two.)

    If my numbers are correct, there is one value of b that gives a coefficient of 0 for the 1/x term and the 1/(x + 1) term (the ones that produce log terms when integrated). The other terms do not produce log terms when integrated.
     
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