When I learned about the properties of integrals, as the one below, for instance, I noticed the preamble of the theorem requires the respective functions to be both bounded and integrable.
If $f$ and $g$ are bounded, integrable functions on $[a, b]$, then so is $f + g$ and
$$ \int_a^b (f(x) + g(x)) dx = \int_a^b f(x) dx + \int_a^b g(x) dx $$
My question is why do the functions have to be bounded? Isn't it enough for them to be integrable? I have not completed my Calculus course yet, so there are certain concepts I don't know about yet, like improper integrals.