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Math Help - Integration-Partial Fractions

  1. #1
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    Integration-Partial Fractions

    <br />
\int {\frac{{dx}}<br />
{{(x^2 - 1)^2 }}} = \int {\frac{{dx}}<br />
{{(x - 1)^2 (x + 1)^2 }}} <br />

    It seems no matter what I try after this I cannot get the answer in the book.
    Which is:

    <br />
\frac{1}<br />
{4}\int {(\frac{{ - 1}}<br />
{{x - 1}}} + \frac{1}<br />
{{(x - 1)^2 }} + \frac{1}<br />
{{x + 1}} + \frac{1}<br />
{{(x + 1)^2 }})dx<br />


    Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by kid funky fried View Post
    <br />
\int {\frac{{dx}}<br />
{{(x^2 - 1)^2 }}} = \int {\frac{{dx}}<br />
{{(x - 1)^2 (x + 1)^2 }}} <br />

    It seems no matter what I try after this I cannot get the answer in the book.
    Which is:

    <br />
\frac{1}<br />
{4}\int {(\frac{{ - 1}}<br />
{{x - 1}}} + \frac{1}<br />
{{(x - 1)^2 }} + \frac{1}<br />
{{x + 1}} + \frac{1}<br />
{{(x + 1)^2 }})dx<br />


    Any help would be greatly appreciated.
    \frac{1}{(x^2-1)^2}=\frac{1}{(x-1)^2(x+1)^2}

    \frac{1}{(x-1)^2(x+1)^2}=\frac{A}{(x-1)}+\frac{B}{(x-1)^2}+\frac{C}{(x+1)}+\frac{D}{(x+1)^2}

    Clearing the fractions gives

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

    First lets get our two freebee values

    let x =1

    1=A(1-1)(x+1)^2+B(1+1)^2+C(1-1)^2(1+1)+D(1-1)^2
    1=b(2)^2 \iff \frac{1}{4}=B

    x=-1 gives

    1=D(-1-1)^2 \iff \frac{1}{4}=D

    Now we know that this is true for ALL VALUES OF X so we can pick some others. I will pick x=0 and x=2 to get a system of equations.

    1=A(0-1)(0+1)^2+\frac{1}{4}(0+1)^2+C(0-1)^2(0+1)+\frac{1}{4}(0-1)^2

    1=-A+\frac{1}{4}+C+\frac{1}{4} \iff \frac{1}{2}=-A+C

    1=A(2-1)(2+1)^2+\frac{1}{4}(2+1)^2+C(2-1)^2(2+1)+\frac{1}{4}(2-1)^2

    1=9A+\frac{9}{4}+3C+\frac{1}{4} \iff -\frac{3}{2}=9A+3C \iff -\frac{1}{2}=3A+C

    Now we need to solve
    3A+C=-\frac{1}{2}
    -A+C=\frac{1}{2}

    subtracing the 2nd from the first gives

    4A=-1 \iff A=-\frac{1}{4}

    then -\left( -\frac{1}{4}\right)+C=\frac{1}{2} \iff C=\frac{1}{4}

    There we go Yeah!!!

    I hope this helps.
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  3. #3
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    Yes it helps alot. I had gotten as far as finding B and D. A and C had me perplexed.
    Thanks, I appreciate it.
    Last edited by kid funky fried; May 3rd 2008 at 06:48 PM.
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