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\problemgrouptitle{Partial Fraction Decomposition}

\begin{problemblock}[2]{Evaluate the given integral.}

\item $\displaystyle\int \frac {x}{x^2-5x+6}\, dx$
\answer{
$-2\ln | x-2 | +3 \ln |x-3|+C$
}

\item $\displaystyle\int \frac {x^3+1}{x^2-9}\, dx$
\answer{
$\displaystyle \frac 12 x^2+\frac{13}3\ln | x+3 | +\frac {14}3 \ln |x-3|+C$
}

\item $\displaystyle\int \frac {3x-1}{\left (x-1\right)^2 (x+2)}\, dx$
\answer{
$\displaystyle \frac{21}{27}\ln | x-1 | -\frac 23 \frac 1{x-1}-\frac {7}9 \ln |x+2|+C$
}

\item $\displaystyle\int \frac {2x+4}{\left (x+1\right) \left(x^2+1\right)}\, dx$
\answer{
$\displaystyle \ln | x+1 | -\frac 12 \ln |x^2+1|+3 \tan ^{-1} x+C$
}

\item $\displaystyle\int \frac {x-9}{x\, \left(x^2+3\right)^2}\, dx$
\answer{
$\displaystyle -\ln | x| +\frac 12 \ln |x^2+3|-\frac 32 \frac 1{x^2+3}+\frac {\sqrt 3}{18} \tan ^{-1} \frac x{\sqrt 3}+ \frac 16 \frac x {x^2+3}+C$
}

\item $\displaystyle\int \frac {x^4+x^2+1}{ \left(x-1\right)^3}\, dx$
\answer{
$\displaystyle\frac {x^2}2+3x+7 \ln | x-1| -\frac 6{x-1} -\frac 32 \frac 1{(x-1)^2}+C$
}

\item $\displaystyle\int \frac {10}{ x^2\, \left(x^2 +5\right)}\, dx$
\answer{
$\displaystyle -\frac {2}x- \frac 2{\sqrt 5} \tan^{-1}\!\left( \frac x {\sqrt 5} \right)+C$
}

\item $\displaystyle\int \frac {x^6+x^3-6}{ x^4+3x^2}\, dx$
\answer{
$\displaystyle \frac {x^3}3-3x+\frac 2x+\frac 12  \ln | x^2+3| +\frac {11}{\sqrt 3}\tan^{-1}\left(\frac{x}{\sqrt 3}\right)+C$
}

\item $\displaystyle\int \frac {x^2+2x+5}{ x^4+4x^2+3}\, dx$
\answer{
$\displaystyle -\frac 12 \ln | x^2+3| -\frac 1{\sqrt 3}\tan^{-1}\! \left (\frac x{\sqrt 3} \right)+\frac 12 \ln |x^2+1 |+2\tan^{-1}\! x+C$
}

\item $\displaystyle\int \frac {2x^2-4x+4}{ x^3-x^2+x-1}\, dx$
\answer{
$\displaystyle \ln | x-1|+\frac 12 \ln |x^2+1 |-3\tan^{-1}\! x+C$
}

\end{problemblock}

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