# Thread: Integration of Rational Functions By Partial Fractions

1. ## 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

2. $

\int {\frac{{3x^2 + 7x + 3}}
{{\left( {x^2 + 1} \right)^2 }}dx} = \int {\frac{{3x^2 + 3}}
{{\left( {x^2 + 1} \right)^2 }}dx + \int {\frac{{7x}}
{{\left( {x^2 + 1} \right)^2 }}dx} } =

$

$
3\int {\frac{1}
{{x^2 + 1}}dx} + 7\int {\frac{x}
{{\left( {x^2 + 1} \right)^2 }}dx}

$

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

$
= \boxed{3\arctan x + \frac{7}
{2} \cdot \frac{1}
{{x^2 + 1}}}

$

I forget the integration constant :P

3. Thanks, but the program I'm using isn't accepting that answer. Are you sure it's right?
Jay

4. Originally Posted by HellaciousD
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

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

Thanks,
Jay