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Math Help - Integration of Rational Functions By Partial Fractions

  1. #1
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    Integration of Rational Functions By Partial Fractions

    Hello all,

    I am trying to integrate (3x^2+7x+3)/(x^2+1)^2
    I am down to the integral of (3/(x^2+1))+ the integral of (7x/(x^2+1)^2)+ the integral of (3/(x^2+1)^2) and I don't really know what to do
    I believe the next step starts with arctan(3x) but then I don't know what to do with the other stuff.

    Thanks,
    Jay
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  2. #2
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    <br /> <br />
\int {\frac{{3x^2  + 7x + 3}}<br />
{{\left( {x^2  + 1} \right)^2 }}dx}  = \int {\frac{{3x^2  + 3}}<br />
{{\left( {x^2  + 1} \right)^2 }}dx + \int {\frac{{7x}}<br />
{{\left( {x^2  + 1} \right)^2 }}dx} }  = <br /> <br />

    <br />
3\int {\frac{1}<br />
{{x^2  + 1}}dx}  + 7\int {\frac{x}<br />
{{\left( {x^2  + 1} \right)^2 }}dx} <br /> <br />

    <br />
\begin{gathered}<br />
  u = x^2  + 1 \Rightarrow du = 2xdx \hfill \\<br />
   \hfill \\<br />
   = 3\arctan x + \frac{7}<br />
{2}\int {\frac{1}<br />
{{u^2 }}du = } 3\arctan x + \frac{7}<br />
{2} \cdot \frac{{u^{ - 2 + 1} }}<br />
{{ - 2 + 1}} \hfill \\ <br />
\end{gathered} <br />

    <br />
 = \boxed{3\arctan x + \frac{7}<br />
{2} \cdot \frac{1}<br />
{{x^2  + 1}}}<br /> <br />

    I forget the integration constant :P
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  3. #3
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    Thanks, but the program I'm using isn't accepting that answer. Are you sure it's right?
    Jay
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  4. #4
    Member Nacho's Avatar
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    Quote Originally Posted by HellaciousD View Post
    Thanks, but the program I'm using isn't accepting that answer.
    Jay
    look that I make a mistake in a sign, before 7/2... should to have a "-"

    sorry
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  5. #5
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    Can someone tell me why the integral of 32dx/(x(x^2+4)^2) doesn't come out to 2ln(abs(x))?

    Thanks,
    Jay
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