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Math Help - [SOLVED] find integral of partial fraction

  1. #1
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    [SOLVED] find integral of partial fraction

    \int \frac{2x-3}{(x-1)^2}dx = \frac{A}{x-1}+\frac{B}{(x-1)^2}

    A=(-1/3)
    B=(8/3)

    I don't know how to find the Ax^2...Please Help.
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  2. #2
    Senior Member mollymcf2009's Avatar
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    see below post
    Last edited by Jameson; March 3rd 2009 at 09:11 PM. Reason: accidental mistake made
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  3. #3
    Member Nacho's Avatar
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    <br />
\begin{gathered}<br />
  \frac{{2x - 3}}<br />
{{\left( {x - 1} \right)^2 }} = \frac{A}<br />
{{x - 1}} + \frac{B}<br />
{{\left( {x - 1} \right)^2 }} \hfill \\<br />
  2x - 3 = A\left( {x - 1} \right) + B \Leftrightarrow 2x - 3 = Ax - A + B \hfill \\ <br />
\end{gathered} <br />

    <br />
 \Rightarrow \left. {\underline {\, <br />
 \begin{gathered}<br />
  A = 2 \hfill \\<br />
   - A + B =  - 3 \hfill \\ <br />
\end{gathered}  \,}}\! \right|  \Rightarrow B = 1 \Rightarrow \frac{{2x - 3}}<br />
{{\left( {x - 1} \right)^2 }} = \frac{2}<br />
{{x - 1}} - \frac{1}<br />
{{\left( {x - 1} \right)^2 }}<br />

    But look at this:

    <br />
\int {\frac{{2x - 3}}<br />
{{\left( {x - 1} \right)^2 }}} dx = \int {\frac{{2x - 2}}<br />
{{x^2  - 2x + 1}}dx - \int {\frac{1}<br />
{{\left( {x - 1} \right)^2 }}} } dx<br />

    To the first integral:

    <br />
\int {\frac{{2x - 2}}<br />
{{x^2  - 2x + 1}}dx}  = \int {\frac{{\left( {x^2  - 2x + 1} \right)^\prime  }}<br />
{{x^2  - 2x + 1}}dx}  = \ln \left( {x^2  - 2x + 1} \right) + k<br />

    The second integral is direct
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  4. #4
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    post about previous post, containing error. see above post.
    Last edited by Jameson; March 3rd 2009 at 09:36 PM.
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  5. #5
    Senior Member mollymcf2009's Avatar
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    contains same error
    Last edited by Jameson; March 3rd 2009 at 09:36 PM.
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  6. #6
    Super Member redsoxfan325's Avatar
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    What Nacho said to do is probably how I'd go about it (especially because some of the other posts are incorrect). BUT, if you have to use partial fractions, then let:

    \frac{2x-3}{(x-1)^2} = \frac{A}{x-1}+\frac{B}{(x-1)^2}

    Multiplying through by
    (x-1)^2 gives you A(x-1)+B = Ax-A+B = (A)x + (-A+B) = 2x-3.

    Set A = 2 and it follows that B=-1.

    Thus, \int \frac{2x-3}{(x-1)^2}\,dx = \int \frac{2}{x-1}\,dx-\int\frac{1}{(x-1)^2}\,dx.

    The first integral is simply 2ln(x-1) = ln((x-1)^2) = ln(x^2-2x+1), which if you note, is the same answer Nacho got using his method.

    The second integral is -\frac{1}{x-1}.

    Thus ln(x^2-2x+1)-(-\frac{1}{x-1}) = ln(x^2-2x+1)+\frac{1}{x-1}.

    \int \frac{2x-3}{(x-1)^2}\,dx = ln(x^2-2x+1)+\frac{1}{x-1} + C
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  7. #7
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    Quote Originally Posted by mollymcf2009 View Post
    Expand your equations:

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

    = A(x-1)^2 + B(x-1)
    I'm not seeing what you're doing here. I think Nacho's work below this post is correctly done. You conclude in your second post in this thread that the original fraction can be replaced by one different fraction - which intuitively doesn't make sense. I think this is an error.
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  8. #8
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    oops. I completely missed that. Thanks guys.
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  9. #9
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    this is what I have when I rearrange it:
    6x=x^2(A+B)+x(2A-2B+C)+(4A-2C)

    I'm having trouble solving for A,B, and C. Wouldn't the x(2A-2B+C) = 6?
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  10. #10
    Super Member redsoxfan325's Avatar
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    You probably meant to post that here.

    But yes, 2A-2B+C=6. Also A+B = 0 and 4A-2C=0.

    Solve the system of equations to find A, B, and C.
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