find the partial fraction decomposition of (x+3)/x^2*(x^2+4)?

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  • Mangal
    Lv 4
    2 months ago

    see image.... .... .... 

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  • a/x + b/x^2 + cx/(x^2 + 4) + d/(x^2 + 4) = (x + 3) / (x^2 * (x^2 + 4))

    a * x * (x^2 + 4) + b * (x^2 + 4) + (cx + d) * x^2 = x + 3

    ax^3 + 4ax + bx^2 + 4b + cx^3 + dx^2 = 0x^3 + 0x^2 + x + 3

    ax^3 + cx^3 = 0x^3

    a + c = 0

    a = -c

    bx^2 + dx^2 = 0x^2

    b + d = 0

    b = -d

    4ax = x

    4a = 1

    a = 1/4

    c = -1/4

    4b = 3

    b = 3/4

    d = -3/4

    (1/4) / x  +  (3/4) / x^2  -  (1/4) * (x + 3) / (x^2 + 4) =>

    (1/4) * (1/x + 3/x^2 - (x + 3) / (x^2 + 4))

    Test

    ((1 * x * (x^2 + 4) + 3 * (x^2 + 4) - (x + 3) * x^2)) / (x^2 * (x^2 + 4)) =>

    (x^3 + 4x + 3x^2 + 12 - x^3 - 3x^2) / (x^2 * (x^2 + 4)) =>

    (4x + 12) / (x^2 * (x^2 + 4))

    (1/4) * (4x + 12) / (x^2 * (x^2 + 4)) =>

    (x + 3) / (x^2 * (x^2 + 4))

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