When dealing with improper integrals I sometimes have to figure out whether or not to use the comparison test. Everything I read says something along the lines of "Also, there will be some integrals that we simply won’t be able to integrate and yet we would still like to know if they converge or diverge."
How do I actually figure out whether or not I can integrate without actually trying to integrate and failing?
In full generality this is an extremely hard problem, in fact it is impossible.
In practice, by learning lots of integration techniques and integration criteria you develop better intuition and have more tools to answer the question. Also, knowing many examples of non-integrable functions helps in quickly spotting more such functions. But, there is not general method.
Typically if you can find an upperbound for your integrand on the region of integration you can then conclude convergence.
In general, if $\int_{a}^{b}f\leq f(x)$ for some $x\in[a,b]$ then $f$ converges in $[a,b]$.
However, this is a very special thing to be able to find upperbounds, so I would suggest learning some elementary and non-elementary integration techniques.
