In Lebesgue's theory of integration, the integral $$ \int_{(1,\infty)} \frac{\sin x}{x^2} \,dx \tag 1 $$ is not defined as $$ \lim_{a\to\infty} \int_{(1,a)} \frac{\sin x}{x^2}\,dx, $$ although one can show that it's equal to that. On the other hand the integral $$ \int_{(0,\infty)} \frac{\sin x} x\,dx \tag 2 $$ does not exist at all in Lebesgue's theory, since if $A = \left\{x\in(0,\infty) : \dfrac{\sin x} x\ge 0\right\}$ then $$ \int_A \frac{\sin x} x\,dx = +\infty \text{ and } \int_{(0,\infty)\,\setminus\, A} \frac{\sin x} x\,dx = -\infty. \tag 3 $$ However, $$ \lim_{a\to\infty} \int_0^a \frac{\sin x} x \, dx \tag 4 $$ exists and is a strictly positive finite number.
I have the impression that
- Some purists refuse to use the term "improper integral" for things like $(1)$, since the positive and negative parts are finite and so it can be defined without first defining an integral over a bounded interval and then taking a limit, and they reserve the term "improper integral" for things like $(2)$ (which is defined by saying it's equal to the limit in $(4)$), where the positive and negative parts are both infinite;
- All (?) elementary calculus texts use the term "improper integral" whenever one integrates over an unbounded set (and also when the function itself is unbounded even if its domain is bounded).
Perhaps these are two extreme positions and others exist between the two extremes. What opinions are out there, on how the terminology ought to be used, and why?
Also, some integrals that are not improper in the purists' sense, such as $$ \int_0^\infty e^{-x} \, dx, $$ seem to be most readily computed by taking a limit like that in $(4)$. Are there provable results to the effect that that is in some sense the simplest way to do it? If one were a purist as described above, what name would one then use for integrals most simply computed in that way?