What is the point of spending so much time studying integrals?
Hi there! A physics student here. For a long time I have seen both sides of integration study: mathematical and practical. I noticed, that most math exams checking basic tricks (substitution, by parts, etc.) automatically expect you supposing existence of an elementary antiderivative of the given integrand. When you have an integral problem you expect that it's made to be solved, because exams have given you exactly this kind of experience. But my point is, that in reality you don't have this beauty in most of the cases. Even the most basic examples, such as, for example, air resistance (dv/dt = k * v^2 + mg, dx/dt = v, you need to integrate twice) is on edge of being solvable analytically. If I complicate it a little bit more, then I can be sure that there is no elementary antiderivative for the given case. You would have to use numerical integration aka approximation.
What is the point then of putting so much practice into things that you mostly will never ever use in the classical form?
I highly respect any opinions, and my point here is not to trashtalk strict math fans, but to have a discussion.