Few confusions about integration There are some confusions I have about integration, and I hope you can help me clearing them up.
1- Why don't we ever pay attention to the domain of the integrand? especially when there are many functions that get integrated and do have discontinuity points / intervals. Do we implicitly integrate the integrand for only the domain of it? where ONLY the function(integrand) is defined? Like, does integration assume only the defined intervals of the integrand? 
2- Why don't we pay attention whether absolute value property should be used or not?Especially when dealing with radicals (could follow the first question). This issue showed up when I tried to solve this integral: Any hints on how to compute this integral?
when 3 users suggested to ignore the domain so that I could avoid the absolute value issue.
3- Comparison properties of definite integrals
Why do we use them if we can already know the value of the integration? They aren't ambiguous or variable so that we would use them. Not to mention that we even use equal or (greater or less) than operator which confuses me more.
 A: *

*We do, or should, pay attention to domains.  If a function is not defined everywhere in an interval, its (Riemann) integral over that interval is not defined.  If there are a finite number of points of the interval where the function is not defined, we may be treat the integral as an improper integral, which may or may not converge.

*I don't know what you mean by "absolute value property". In the answer you linked to, Yves Daoust commented that the domain must be considered - he just didn't bother doing it.

*Comparisons are useful when you don't know the actual value of the integral.  It may be difficult to compute, or even impossible to find in closed form.

A: What do you mean by integral? That's where we should start from. For it seems that you're taking integral to be how it is usually defined in elementary calculus texts, or calculus for non-majors, namely the integral of a function is defined as the function whose derivative is the integrand. Usually this is then followed by the non-uniqueness of such 'integrals', but since they all only differ by a constant, so they continue, we can regard the integral as well defined up to a constant, and so on.
But as you advance in analysis you get to have a well-defined meaning of the term integral, which assigns to (some types of) real-valued functions a unique real number satisfying some properties. In this definition, it is even impossible to conceive of integral without considering domains of the integrand, for this is necessary to define the integral. Thus, we integrate functions over well-specified domains. I believe this answers your first two questions.
As for your last, I don't fully understand your question, so you may say so if I'm on track or not. But we compare integrals for the same reason we compare series, for example. We might want, not to calculate an integral exactly, but only to estimate its value; for different reasons -- it might be very hard to do directly, for example, or even if it isn't, it might be a hopeless combination of irrationals that we would have to approximate after all. Then it's clear why we might have to compare to other integrals that we already understand sufficiently well. Even from the purely theoretical perspective, estimates help in solving or proving some otherwise intractable problems or results -- isn't that the power of analysis?
I hope this helps.
