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\title{\vspace*{-1.5in}\textbf{\S5.7: Partial Fractions Decomposition}\\\normalsize (Thanks to Faan Tone Liu)}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{January 29, 2018}
\rhead{Partial Fractions Decomposition}
\vspace{-.5in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item Use this method to integrate rational functions.
\item Degree of numerator must be lower than degree of denominator. If needed, start with long division.
\item Factor the denominator
\item Solve the decomposition according to the right form:
\begin{itemize}
\item Linear Factors: \quad$\displaystyle \frac{5x+3}{(x+1)(x+4)} = \frac{A}{x+1} + \frac{B}{x+4}$\\[12pt]
\item  Repeated Linear Factors: \quad$\displaystyle \frac{2x-4}{(x-2)(x+3)^2} = \frac{A}{x-2} + \frac{B}{x+3} + \frac{C}{(x+3)^2}$\\[12pt]
\item  Irreducible Quadratic Factors: \quad$\displaystyle \frac{2x^2-3x-1}{(x-1)(x^2+9)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+9}$\\[12pt]
\item Mixtures are possible!
\end{itemize}
\item Make sure you remember how to integrate the ``outputs'' of the partial fractions decomposition. For example:
\[
\frac{5}{2x+1},\,\, \frac{3}{(x+4)^2},\,\, \frac{2x}{x^2 + 9},\,\, \frac{4}{x^2 + 25},\,\, \text{or}\phantom{,}\,\, \frac{3x+1}{x^2+16}. 
\]
\item Other notes and tips:
\end{itemize}\newpage


\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item $\displaystyle \int \frac{2x^2+3x-3}{x^2-x}\,dx$\\[12pt]
First, use long division to re-write the integrand so the fraction involved has a numerator with a lower degree than the denominator.\vskip1.2in

\noindent
The form of the decomposition of $\frac{5x-3}{x(x-1)}$ is:\vskip2cm

\noindent
Here are two methods to find $A$ and $B$ in the partial fractions decomposition:\\[12pt]
\hspace*{1.45cm}\begin{tabular}{c|c}
Method I: Equate Coefficients & Method II: Judicious Substitution\\\hline
&\\[1.75in]
\end{tabular}\vskip1cm

\noindent
Now, $\displaystyle \int \frac{2x^2+3x-3}{x^2-x}\,dx = $
\newpage

\item $\displaystyle \int \frac{x^2+x-5}{(x-2)(x-1)^2}\,dx$
\newpage

\item $\displaystyle\int \frac{10}{(x+1)(x^2+9)}\,dx$

\end{enumerate}
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