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Problem resolving an Integral - Partial Fractions

  1. Feb 3, 2013 #1
    1. So, i have the next integrand...



    2. [itex]\int \frac{1}{(x-1)^2(x+1)^2}\,dx[/itex]



    3. I proceeded by resolving it by partial fraction and i came up with the next...

    [itex]\int \frac{1}{((x-1)^2)((x+1)^2)}\,dx = \int \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{C}{(x+1)} + \frac{D}{(x+1)^2}\,dx[/itex]

    The thing is that after doing all the calculus i came up with this...

    [itex]A + C = 0[/itex]
    [itex]A + B - C + D = 0[/itex]
    [itex]-A + 2B - C -2D = 0[/itex]
    [itex]A + B + C + D = 1[/itex]

    After this i dont know how to preceed i mean i dont know how to resolve the equation whit 4 variables....

    Thanks, and very sorry for my english...
     
    Last edited: Feb 3, 2013
  2. jcsd
  3. Feb 3, 2013 #2

    SammyS

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    Use [itex]\ \ A + C = 0\ \ [/itex] with [itex]\ \ -A + 2B - C -2D = 0\ \ [/itex]

    to get [itex]\ \ 2B -2D = 0\ .[/itex]

    Similarly, use [itex]\ \ A + C = 0\ \ [/itex] with [itex]\ \ A + B + C + D = 1[/itex]

    to get [itex]\ \ B + D = 1\ .[/itex]

    Use those two equations to solve for B & D .

    Put the results for B & D into the first two equations to get A & C .
     
  4. Feb 3, 2013 #3

    SammyS

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    As an alternative to the above method, consider the following. (This method was also discussed in the thread: Partial Fractions)

    I assume that you got your 4 equations for A, B, C, and D by equating coefficients for powers of x in the following equation.

    [itex]1=A(x-1)(x+1)^2+B(x+1)^2+C(x-1)^2)(x+1)+D(x-1)^2\ .[/itex]

    Your can quickly solve for B, by letting x = 1 .

    Similarly, you can solve for D, by letting x = -1 .

    After that it's not so difficult to find A & C .
     
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