The first “open question” that I asked was why one can’t integrate things of the form cos(x^n) or sin(x^n) when n is larger than 1. This wasn’t a serious question, just something to think about. The way that I came to conclusion that it couldn’t be done is realizing that e^(x^n) can’t be done.

By Euler’s extension of complex exponentials, e^øi = cosø + i sinø, one can see that ∫cosø^2 dø would be the real part of (∫e^(iø^2) dø). And since e^(x^2) can’t be done (again, with elementary functions) nor can cosø^2 be done.

The other not so serious question was to explain the, ahem, “graph”. It was a graph of two cycloids with parabolas in the middle. It was done as a piece-wise function so that only part of the parabola’s above the cycloids could be seen. This “graph” ended up looking like something interesting.