%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Indefinite Integration ">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:06:08 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="begin" HREF="DM7-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="end" HREF="DM7-9.tex">}
%TCIDATA{<LINK REL="next" HREF="DM7-5.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM7.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM7-3.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{Indefinite Integration\label{Indefinite integral}}

An \emph{antiderivative} of a \label{Antiderivative} 
\index{Antiderivative}function $f(x)$ is any function $g(x)$ whose
derivative is $f(x)$. If $g(x)$ is an antiderivative of $f(x)$, then $g(x)+C$
is another antiderivative. In fact, every antiderivative is of the form $%
g(x)+C$ for some constant $C$.

The \emph{indefinite integral} of $f(x)$ is the 
\index{Indefinite integral} 
\index{Integral!indefinite}family of all antiderivatives of $f(x)$ and is
denoted $\int f(x)\,dx$.

\begin{quote}
$\blacktriangleright $ To evaluate an indefinite integral
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point anywhere in the expression.

\item[2.]  Choose \textsf{Evaluate}, press \textsc{ctrl + e}, or click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{evaluate.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Evaluate.wmf';file-properties "XNPEU";}}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\dint \left( 2x^{2}+3x+5\right) \,dx=\allowbreak \frac{2}{3}x^{3}+\frac{3}{2%
}x^{2}+5x$
\end{quotation}

\ 

\label{Piecewise-defined function}You can evaluate indefinite integrals of
piecewise-defined functions. The result will be a piecewise-defined function.

\begin{example}
Make the definition 
\[
f(x)=\left\{ 
\begin{array}{lll}
x & if & x<0 \\ 
3x^{2} & if & x\geq 0
\end{array}
\right. 
\]
Then \textsf{Evaluate} will return the following piecewise-defined function.
\end{example}

\[
\int f(x)dx=\allowbreak \left\{ 
\begin{array}{ccc}
\frac{1}{2}x^{2} & \func{if} & x\leq 0 \\ 
x^{3} & \func{if} & 0<x
\end{array}
\right. 
\]

\ 

\subsection{\ Constant of Integration}

\label{Constant of integration}The system does not automatically return the
constant of integration--the ``arbitrary constant,'' so you \label%
{Arbitrary constant} 
\index{Constant of integration}must remain alert and add the constant when
needed. Simply enter ``$+\ C\,$'' to change from 
\[
\int \left( 2x^{2}+3x+5\right) \,dx=\allowbreak 
\frac{2}{3}x^{3}+\frac{3}{2}x^{2}+5x 
\]
to 
\[
\dint \left( 2x^{2}+3x+5\right) \,dx=\allowbreak \frac{2}{3}x^{3}+\frac{3}{2}%
x^{2}+5x+C 
\]
Such constants are needed, for example, if you have a formula for
acceleration and you wish to find an expression for velocity.

\subsection{\ Interpreting an Expression\label{Interpreting expressions}}

If the insertion point is placed in a mathematical expression, \textsc{ctrl
+ ?} gives \textsl{Scientific Notebook}'s interpretation 
\index{Interpreting expressions} of that expression.

\begin{quote}
$\blacktriangleright $ \textsc{ctrl + ?}
\end{quote}

\begin{quotation}
$\dint ax^{3}=\allowbreak \dint ax^{3}\,d?$\qquad

$\dint x^{3}a=\allowbreak \dint x^{3}a\,d?$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\dint ax^{3}=\allowbreak 
\frac{1}{4}ax^{4}$\qquad

$\dint x^{3}a=\allowbreak \frac{1}{4}ax^{4}$
\end{quotation}

If $f$ is not defined as a function, then it is treated like any other
letter.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\dint f=\allowbreak \frac{1}{2}f^{2}$\qquad

$\dint f(x)dx=\allowbreak \frac{1}{2}fx^{2}$
\end{quotation}

In these expressions, $f$ behaves the same as any other variable, and $f(x)$
is interpreted as simply the product of $f$ and $x$.

\subsection{\ Sequences of Operations\label{Sequences of operations}}

Sometimes \textsl{Scientific Notebook} returns answers that are not in the
form you need. You can take advantage of items such as \textsf{Simplify}, 
\textsf{Combine}, \textsf{Expand}, and \textsf{Factor} under the \textsf{%
Maple} menu to rewrite these answers. Here is an example that illustrates
their use.

\begin{example}
Place the insertion point in the expression $\int e^{ax}\cos bx\,dx$ and
choose \textsf{Evaluate}. Without moving the insertion point, choose \textsf{%
Factor}. The result is now in a nice form. 
\begin{eqnarray*}
\int e^{ax}\cos bx\,dx &=&\allowbreak \frac{a}{a^{2}+b^{2}}e^{ax}\cos bx+%
\frac{b}{a^{2}+b^{2}}e^{ax}\sin bx \\
&=&\allowbreak e^{ax}\frac{a\cos bx+b\sin bx}{a^{2}+b^{2}}
\end{eqnarray*}
\end{example}

The integral $\int 2^{x}\cos bxdx$ looks just as straightforward but
requires several extra steps to get the answer into an equivalent final
form. (See the \hyperref{exercises}{}{}{DM7-9.tex#Exercises} at the end of
this chapter.) Simplification of algebraic expressions is more subtle than
it might at first appear. Sometimes you just have to experiment with the
menu items \textsf{Simplify}, \textsf{Combine}, \textsf{Expand}, and \textsf{%
Factor} to get expressions into a manageable form. This interaction with the
system is a natural part of the machine-human interface.

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Evaluating Calculus Expressions}{}{}{DM7-1.tex#Calculus
expressions}

\item  \hyperref{Limits}{}{}{DM7-2.tex#Limit}

\item  \hyperref{Differentiation}{}{}{DM7-3.tex#Derivative}

\item  \hyperref{Methods of Integration}{}{}{DM7-5.tex#Methods of integration%
}

\item  \hyperref{Definite Integrals}{}{}{DM7-6.tex#Definite integral}

\item  \hyperref{Exercises and Solutions}{}{}{DM7-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.125pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index} {\small Index entries: \hyperref{Antiderivative}{}{}{%
DMIndex.tex#Antiderivative}, \hyperref{Constant of integration}{}{}{%
DMIndex.tex#Constant of integration}, \hyperref{Indefinite integral}{}{}{%
DMIndex.tex#Indefinite integral}, \hyperref{Sequences of operations}{}{}{%
DMIndex.tex#Sequences of operations}}

\end{document}
