Improper Riemann Integral

Question

I am supposed to solve the following question:

For what function classes it makes sense to talk about the Improper Riemann Integral?

I know that we can talk now about bounded functions defined on unbounded interval or unbounded functions defined on bounded interval.

But is there any more specific answer?

Answer 1

For the first case, you can assume that the interval is in the form $[a,+\infty)$ for some $a$ (the $(-\infty,a]$ case us similar). Then the integral of $f$ on $[a,+\infty)$ is defined as $$\int_a^{+\infty} f := \lim_{b \to +\infty } \int_a^b f$$ And if $f$ is "unbounded at $a$", then the integral on $(a,b]$ is defined as $$\int_a^b f = \lim_{x \to a+0} \int_x^b f$$ What do we need for these definitions to "make sense"?