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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Integration of Rational Functions by Partial Fractions}

A rational function can be integrated by expressing it as a sum of smaller
fractions. Consider a rational function 
\[
f\left( x\right) =\frac{P\left( x\right) }{Q\left( x\right) } 
\]
where $P$ and $Q$ are polynomials. It is possible to express $f$ as a sum of
smaller fractions if the degree of $P$ is less than the degree of $Q$. Such
a rational function is called \textbf{proper}. If $f$ is \textbf{improper} ($%
\deg \left( P\right) \geq \deg \left( Q\right) $), you must first divide $P$
by $Q$ until you have a remainder $R\left( x\right) $ such that $\deg \left(
R\right) <\deg \left( Q\right) $. If $f$ is improper, the result after
division is 
\[
f\left( x\right) =\frac{P\left( x\right) }{Q\left( x\right) }=S\left(
x\right) +\frac{R\left( x\right) }{Q\left( x\right) } 
\]
where $S$ and $R$ are also polynomials.

\subsubsection{Partial Fractions\label{partial fractions}}

Suppose that $f$ is an improper rational function. The first step in solving
is to perform long division to obtain the equation 
\[
f\left( x\right) =S\left( x\right) +\frac{R\left( x\right) }{Q\left(
x\right) } 
\]
The next step is to factor $Q\left( x\right) $ as far as possible. Any
polynomial $Q$ can be expressed as a product of linear factors (of the form $%
ax+b$) and irreducible quadratic factors (of the form $ax^{2}+bx+c$, where $%
b^{2}-4ac<0$). The next step is to express the proper rational function $%
\frac{R\left( x\right) }{Q\left( x\right) }$ as a sum of \textbf{partial
fractions} of the form 
\[
\frac{A}{\left( ax+b\right) ^{i}}\text{ or }\frac{Ax+B}{\left(
ax^{2}+bx+c\right) ^{j}} 
\]

\begin{enumerate}
\item[Case 1]  \textbf{The denominator }$Q\left( x\right) $\textbf{\ is a
product of distinct linear factors. }
\end{enumerate}

This means that $Q$ can be expressed as follows: 
\[
Q\left( x\right) =\left( a_{1}x+b_{1}\right) \left( a_{2}x+b_{2}\right)
\cdots \left( a_{k}x+b_{k}\right) 
\]
where no factor is repeated. Then there exist constants $%
A_{1},A_{2},...,A_{k}$ such that 
\begin{equation}
\frac{R\left( x\right) }{Q\left( x\right) }=\frac{A_{1}}{a_{1}x+b_{1}}+\frac{%
A_{2}}{a_{2}x+b_{2}}+...+\frac{A_{k}}{a_{k}x+b_{k}}  \tag{1}  \label{1}
\end{equation}
You can solve for these constants by multiplying both sides of the equation
by $Q\left( x\right) $ and collecting like terms on the right-hand side of
the equation. You then equate the coefficients of like terms on each side of
the equation, and solve the resulting system of equations for the constants.

\begin{enumerate}
\item[Case 2]  $Q\left( x\right) $\textbf{\ is a product of linear factors,
some of which are repeated.}
\end{enumerate}

Suppose that the first linear factor $a_{1}x+b_{1}$ occurs $r$ times in the
factorization of $Q\left( x\right) $. Then instead of the term $\frac{A_{1}}{%
a_{1}x+b_{1}}$ in \hyperref{(1)}{}{}{1} , you would write 
\begin{equation}
\frac{B_{1}}{a_{1}x+b_{1}}+\frac{B_{2}}{\left( a_{1}x+b_{1}\right) ^{2}}+...+%
\frac{B_{r}}{\left( a_{1}x+b_{1}\right) ^{r}}  \tag{2}  \label{2}
\end{equation}
where $B_{1},...,B_{r}$ are constants, and then solve for the constants as
in Case 1.

\begin{enumerate}
\item[Case 3]  $Q\left( x\right) $\textbf{\ contains irreducible quadratic
factors, none of which are repeated.}
\end{enumerate}

If the factorization of $Q\left( x\right) $ contains the factor $ax^{2}+bx+c$%
, where $b^{2}-4ac<0$, then, in addition to the partial fractions in 
\hyperref{\hyperref{(1)}{}{}{1}}{}{}{1} and \hyperref{(2)}{}{}{2}, the
expression $\frac{R\left( x\right) }{Q\left( x\right) }$ will have a term of
the form 
\begin{equation}
\frac{Ax+B}{a^{2}x+bx+c}  \tag{3}  \label{3}
\end{equation}
where $A$ and $B$ are constants. The term in \hyperref{(3)}{}{}{3} can be
integrated by completing the square and using the formula 
\begin{equation}
\int \frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan ^{-1}\left( \frac{x}{a}\right) +C
\tag{4\QTSN{label}{4}}  \label{4}
\end{equation}

\begin{enumerate}
\item[Case 4]  $Q\left( x\right) $\textbf{\ contains a repeated irreducible
quadratic factor.}
\end{enumerate}

If the factorization of $Q\left( x\right) $ contains the factor $\left(
ax^{2}+bx+c\right) ^{r}$, where $b^{2}-4ac<0$, then the sum 
\begin{equation}
\frac{C_{1}x+D_{1}}{a^{2}x+bx+c}+\frac{C_{2}x+D_{2}}{\left(
a^{2}x+bx+c\right) ^{2}}+...+\frac{C_{r}x+D_{r}}{\left( a^{2}x+bx+c\right)
^{r}}  \tag{5}  \label{5}
\end{equation}
occurs in the partial fraction decomposition of $\dfrac{R\left( x\right) }{%
Q\left( x\right) }$. Each of the terms in \hyperref{(5)}{}{}{5} can be
integrated by completing the square and using equation \hyperref{(4)}{}{}{4}.

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Factoring polynomials}{}{}{%
../../Help/DM3-1.tex#Polynomials, factor}}

{\small \hyperref{Method of partial fractions}{}{}{%
../../Help/DM7-5.tex#Partial fractions}}

{\small \hyperref{Partial fractions}{}{}{../../Help/DM3-1.tex#Partial
fractions}}

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