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Math Help - Partial fraction decomposition

  1. #1
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    Partial fraction decomposition

    How do you solve: 1 / x^2(x^2+2x+2)

    I put it in the form:
    A/x + B/x^2 + (Cx+D)/(x^2+2x+2) which can be solved by completing the square

    A/x + B/x^2 + (Cx+D)/[(x+1)^2 + 1] but now I'm stuck... Please help.
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  2. #2
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    Quote Originally Posted by fanban View Post
    How do you solve: 1 / x^2(x^2+2x+2)

    I put it in the form:
    A/x + B/x^2 + (Cx+D)/(x^2+2x+2) which can be solved by completing the square

    A/x + B/x^2 + (Cx+D)/[(x+1)^2 + 1] but now I'm stuck... Please help.
    \frac{1}{x^2(x^2+2x+2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+2x+2}

    1 = Ax(x^2+2x+2) + B(x^2+2x+2) + (Cx+D)x^2

    1 = (A+C)x^3 + (2A+B+D)x^2 + (2A+2B)x + 2B

    equating coefficients ...

    A+C = 0

    2A+B+D = 0

    2A+2B = 0

    2B = 1

    B = \frac{1}{2}

    A = -\frac{1}{2}

    C = \frac{1}{2}

    D = \frac{1}{2}


    \frac{1}{x^2(x^2+2x+2)} = -\frac{1}{2}\left(\frac{1}{x} - \frac{1}{x^2} - \frac{x+1}{x^2+2x+2}\right)
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