Why are certain functions not elementary integrable in the sense of real analysis of one variable, e.g. using the techniques of real analysis of 1 variable? For instance $\nexists \int e^{-x^2}dx$ $\nexists \int \frac{\sin x}{x}dx$ $\nexists \int e^x \tan x dx$.
Of course if one tries to find the definite integral there's no solution in terms of elementary functions but using the Lebesgue theory would be able to find it. So, starting from a dummy level and increasing the abstraction to describe this fact, how are functions that cannot be elementary integrated classified and why? Are there any tests to check whether a function can be elementary integrated or not?
