%Indefinite Integrals
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\Large
Antiderivatives
\footnote{Copyright \copyright 2005 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}
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\begin{enumerate}
\item Enter the following sequence commands and press Enter.
This is an example of a {\em symbolic} computation.
\begin{enumerate}
\item \verb&syms x &
\item \verb&int(x^2)&
\item \verb&diff(ans) &
\item Explain exactly what happened.
\end{enumerate}
\item Repeat steps (b)-(c) for the more complicated function:
$$
\frac{x}{(x - 1)(x + 2)(x^2 - 1)(x + 1)}
$$
(Type: \verb$ int(x / ((x-1)*(x+2)*(x^2-1)*(x+1))) $ for command(b))\\
Then enter the commands: \verb& simplify(ans) & and \verb& pretty(ans) &
\item It is a fact there not every function has an antiderivative which can be
written as a combination of the usual (elementary) functions. Sometimes
\textsc{Matlab} will give a warning when it encouters such a function.
Repeat the above sequence for the following functions:
\begin{enumerate}
\item $\ln(\sin(x^2))$ $\quad$ type: \verb& int(log(sin(x^2))) &
\item $\exp(1+3x+x^5)$ $\quad$ type: \verb& int(exp(1+3*x+x^5)) &
\end{enumerate}
\item Sometimes \textsc{Matlab} will give an answer that involves another problem,
such as solving an equation. Try:\\
$(1 + 3x + x^5)^{-1}$ $\quad$
type: \verb& int(1/(1 + 3*x + x^5)) &
\item Prepare a brief (\verb$< $1 page) written report, including answers to
the questions. Do {\bf not} get a printout. Use standard
mathematical notation and terminology.
Writing quality will play a part in the grade.
\end{enumerate}
\vfill
\noindent
\textsf{This assignment introduces the command for symbolic integration or
antiderivatives. \textsc{Matlab} can do some fairly
sophisticated antiderivatives, but it is not able to find an antiderivative
for some functions. This is not \textsc{Matlab}'s fault, rather it is a
fundamental fact that not all functions have an antiderivatives that are
elementary functions (combination of polynomial, exponential, trig., etc.). }
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