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Math Help - Integratin by Partial Fractions

  1. #1
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    Integratin by Partial Fractions



    Here is what I've done so far...

    (x^3-4x^2-16)/((x^2)(x-4))=A/x+B/x^2+C/(x-4)

    so

    x^3-4x^2-16=A(x^2)(x-4)+B(x)(x-4)+C(x)(x^2)

    = A(x^3-4x^2)+B(x^2-4x)+C(x^3)

    = (A+C)(x^3)+(-4A+B)(x^2)+(-4B)(x)

    Now we have to find what the coefficients are equal to, so

    A+C=1
    -4A+B=-4
    B=0

    A=1
    B=0
    C=0

    So now I should be able to take the integral of 1/x+0/x^2+0/(x-4)

    This equals ln(x) on the interval [7,8], so my answer should be ln(8)-ln(7).

    This is apparently the wrong answer, so I must have made a mistake somewhere, but I'm not seeing it.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kevin100 View Post


    Here is what I've done so far...

    (x^3-4x^2-16)/((x^2)(x-4))=A/x+B/x^2+C/(x-4)

    so

    x^3-4x^2-16=A(x^2)(x-4)+B(x)(x-4)+C(x)(x^2)

    = A(x^3-4x^2)+B(x^2-4x)+C(x^3)

    = (A+C)(x^3)+(-4A+B)(x^2)+(-4B)(x)

    Now we have to find what the coefficients are equal to, so

    A+C=1
    -4A+B=-4
    B=0

    A=1
    B=0
    C=0

    So now I should be able to take the integral of 1/x+0/x^2+0/(x-4)

    This equals ln(x) on the interval [7,8], so my answer should be ln(8)-ln(7).

    This is apparently the wrong answer, so I must have made a mistake somewhere, but I'm not seeing it.
    if the degree of the numerator is greater than or equal to the degree of the denominator, we must divide before doing partial fractions.

    the highest power in the top is the same as that of the bottom. thus we must first divide the numerator by the denominator by long division of polynomials, or alternatively, note that

    \frac {x^3 - 4x^2 - 16}{x^3 - 4x^2} = \frac {x^3 - 4x^2}{x^3 - 4x^2} - \frac {16}{x^3 - 4x^2} = 1 - \frac {16}{x^3 - 4x^2}

    so your integral is \int_7^8 \bigg( 1 - \frac {16}{x^3 - 4x^2} \bigg)~dx

    integrating 1 is easy, use partial fractions for the latter part
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  3. #3
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    I'm still not getting it. My work doesn't seem to make much sense so I wont bother typing it out.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kevin100 View Post
    I'm still not getting it. My work doesn't seem to make much sense so I wont bother typing it out.
    \frac {16}{x^2(x - 4)} = \frac Ax + \frac B{x^2} + \frac C{x - 4}

    \Rightarrow 16 = Ax(x - 4) + B(x - 4) + Cx^2 ...............(1)

    plugging in x = 0 in (1) we get \boxed{B = -4}

    plugging in x = 4 in (1) we get \boxed{C = 1}

    plugging in x=1,~B=-4, and C = 1 in (1) we get \boxed{A = -1}
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  5. #5
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    Krizalid's Avatar
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    Here.
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