When writing the integral sign $\int$, how does one know what integral is being discussed? We have the Lebesgue integral and the Riemann Integral. Generally, must the integral sign $\int$ refer to one or the other exclusively or does it depend on the integrand? Can someone provide intuition to this concept?
For example, in college-level Calculus courses, what integral is actually being used here?

There are only two situations where it could be unclear:


*

*The integral is on a compact interval, $f$ is not Riemann integrable and is Lebesgue integrable;

*The integral is not on a compact interval, $f$ is improperly Riemann integrable and not Lebesgue integrable. 


(Note that e.g. $\int_0^1 \ln(x) dx$ is "not on a compact interval" in this sense.)
The first situation is really not a problem, because either:


*

*the discontinuities of $f$ have positive measure, in which case you're dealing with a Lebesgue integral;

*or it doesn't matter, because both exist and are the same.


The second situation requires careful attention to context. For instance, one should pay careful attention to properly understand what "$\int_1^\infty \frac{\sin(x)}{x} dx$ exists" means.
