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\title{Math 1a.  Lecture 5 \\
Additional Techniques of Integration  \\
(Partial Fractions)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To be able to evaluate definite and indefinite integrals using partial fractions.


\end{itemize} 



\section{Integration by Partial Fractions}

Rational functions can be written as the sum of functions of the form
\begin{itemize}

\item
$\displaystyle \frac{A}{(ax + b)^n}$ 

\item
$\displaystyle \frac{Ax + B}{(ax^2 + bx + c)^n}$ 


\end{itemize}
For example, 
$$
\int \frac{2}{1 - x^2} \, dx = \int \frac{dx}{1 +x} + \frac{dx}{1 - x}
=
\ln|1 + x| - \ln|1 - x| +C.
$$
This is called a {\em partial fraction decomposition}.



\section{Key Examples}


\begin{itemize}


\item 
$\displaystyle \int \frac{x^3}{1 + x^2} \, dx$

\item 
$\displaystyle \int \frac{2 + 5x + 3x^2 + 3x^3}{x(1 + x^2)(x + 2)} \, dx
=
\int \frac{dx}{x} + \int \frac{dx}{1 + x^2} + \int \frac{2 \, dx}{x + 2}
$

\end{itemize}
Show students how to find the partial fraction decomposition.

\section{Notes on the Worksheet}

\begin{enumerate}

\item 
$\displaystyle \int \frac{5x + 7}{(x+1)(x + 2)} \, dx$

\item 
$\displaystyle \int \frac{x^2 + 3x -1}{x(x + 1)(x -2)} \, dx$

\item 
$\displaystyle \int \frac{x^2  -1}{x(x^2 + 4)} \, dx$

\item 
$\displaystyle \int \frac{3x^2 + 7x + 5}{(x + 1)(x^2 + 2x + 2)} \, dx$

\item 
$\displaystyle \int \frac{4x^2  - 3x + 2}{x(2x -1)^2} \, dx$

{\em Hint}: Let
$\displaystyle \frac{4x^2  - 3x + 2}{x(2x -1)^2} = \frac{A}{x} + \frac{B}{2x -1} + \frac{C}{(2x - 1)^2}$.

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

\end{enumerate}



\section*{References}

\begin{itemize}

\item
\S 5.7 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{February 15, 2006}



\end{document}

