I have to ask . . .
Are those really ?
We can integrate: . . . but not
I know we can't integrate and according to Soroban, we can't integrate , but how do we know that for sure? What's the proof that we can't integrate those functions by hand?
This totally my own opinion: Your confusion is understandable and it comes from the very sad conflating of the words integral and antiderivative. They are not the same. An integral is a number, quite often gotten by way of an antiderivative using the fundamental theorem of integral calculus. An antiderivative is just what is says it is.
Of the antiderivative of does exits but we would the series representation for to get it. Therefore, it is proper to say that no elementary representation of the antiderivative of exist in the Calculus II sense of the term.
Now, my TI-92 comes up with:
and the Integrator comes up with
All of these solutions are correct, despite how it might look. The point is that we are doing indefinite integration, so any solution that differs from another by only a constant are all correct. If you spend the time (or just plug it through on your calculator) you will find that all three solutions differ from each other by some constant. (Neither the TI-92 nor the Integrator remind you to add the arbitrary constant on the end.)
I couldn't find reference to Lommel in wiki. Perhaps, that would be a good MathHelpWiki for someone to take on?.
One should be able to use topics from advanced calc to prove sin(x^3) in not integrable by elementary means. Maybe Dirichlet test or something.
It is continuous and differentiable.
I may have to delve into it some more.