\input epsf
\nopagenumbers
\magnification=\magstep1

\centerline{\sevenrm Compute an antiderivative that machines can't
compute! Compute an antiderivative that machines can't compute!
Compute an antiderivative that machines can't compute!}

\smallskip

\noindent {\bf Problem statement} Many students now own calculators
that can find antiderivatives symbolically. This problem is written in
December 2007, and reports on the capability of some of these devices
and some other programs.

\medskip

\noindent a) Find $\int \sqrt{1+\sqrt{x}}\,dx$.

\medskip

\noindent {\bf Hint} Any one of several substitutions will do. You can
try ``the center of the trouble'': $u=\sqrt{1+\sqrt{x}}$ and describe
$dx$ in terms of $u$ and $du$. 

\medskip

\noindent b) Find $\int \sqrt{1+\sqrt{1+\sqrt{x}}}\,dx$.

\medskip

\noindent {\bf Hint and comment} This can be done {\it by hand}. 
Most hand-held devices can't do it. 

\medskip

\noindent c) Describe a strategy which will find $\int
\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{x}}}}\,dx$, explain why your strategy will
be successful, and explain why you would rather not carry out a
detailed implementation of the strategy. 

\medskip

\noindent {\bf Comment} The latest version of the big and powerful
symbolic manipulation program {\tt Maple}, available on most Rutgers
systems, is unable to find this antiderivative (with the default
settings for the program). {\tt Maple}'s major competitor,
{\tt Mathematica}, considers the integral and returns

\smallskip

{\obeylines \obeyspaces \tt
Mathematica could not find a formula for your integral.~Most likely
this means that no formula exists.

}

\smallskip

\noindent which is amusing. 

\smallskip

\centerline{\sevenrm Compute an antiderivative that machines can't
compute! Compute an antiderivative that machines can't compute!
Compute an antiderivative that machines can't compute!}

\vfil\eject\end