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Math Help - Integration by Partial Fractions

  1. #1
    Junior Member qspeechc's Avatar
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    Integration by Partial Fractions

    This problem is taken from Stewart, A58, no.46.

    This is under the section: Integration of Rational Fuctions by Partial Fractions.

    If f is a quadratic function such that f(0) = 1, and
    \intf(x).dx/[x^2.(x+1)^3]
    is a rational function, find the value of f'(0).

    Sorry, but I am having trouble putting it into latex.Any help is much appreciated.
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  2. #2
    Super Member wingless's Avatar
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    f(x) is a quadratic function. We can say that f(x) = a x^2 + bx + c
    If f(0) = 1, we find that c = 1
    So we can rewrite f(x) as ax^2 + bx + 1

    We want to find f'(0).
    f(x) = ax^2 + bx + 1
    f'(x) = 2ax + b
    f'(0) = b
    That means we are looking for b.

    \int \frac{ax^2 + bx + 1}{x^2(x+1)^3}dx

    You can expand it and then integrate it using partial fraction expansions. It's a long work and a real pain to write it here. I assume that you find it correctly. It is,

    \frac{b-a-1}{2(x+1)^2}-\frac{1}{x}+\frac{b-2}{x+1}+(b-3)\ln x + (b-3)\ln (x+1)

    \frac{x^2(2b - 6) + x(3b - a - 9 ) - 2}{2x(x+a)^2}+(b-3)\ln x + (b-3)\ln (x+1)

    b = 3 removes the logarithm functions and leaves us a rational function
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  3. #3
    Junior Member qspeechc's Avatar
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    Aha! That's genius! I got up to that last line of latex, then didn't know what to do. I agree it is a lot of working out. Too much, in fact. I wonder if there is a simpler method of working it out.
    Thank-you, I really never saw that.
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