But this has nothing to do with constructivism. Even if you only allow constructive definitions and proofs, there is still the world of a difference between the definition of an integral and the result you get from evaluating it.

Yeah, sure, in theory you can represent an integral as a function that takes another function and two boundary points and returns a value...

But first, it may not be possible to determine the value of the integral exactly because there is no known method of doing so (the Risch algorithm, apart from it basically being so complex that it's implemented almost nowhere fully, only works for elementary functions!).

And second, if integrals are "just functions", you lose the ability to manipulate them according to known theorems, e.g. additivity, triangle inequality, Cauchy-Schwarz, convergence theorems, ...

So yeah, here's where I get the feeling that some people should do some more maths and spend good parts of their days proving theorems and playing with definitions before they start complaining about how dumb its language is.