%Indefinite Integrals 
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%Math263B

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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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