In calculus, which questions can the naive ask that the learned cannot answer? Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I know. (For now, let's just assume this means the basic topics included in the staid and stagnant conventional first-year calculus course.)
What are


*

*the most prominent and 

*the most readily comprehensible


questions that can be understood by those who know the concepts taught in first-year calculus and whose solutions are unknown?
I'm not looking for problems that people who know only first-year calculus can solve, but only for questions that they can understand.  It would be acceptable to include questions that can be understood only in a somewhat less than logically rigorous way by students at that level.

For what number $k$ is $\int\left(\sin(x^2) + k \cos(x^2)\right)\ dx$ an elementary function?
Hint: $k$ is rational.
