Excellent question. The first thing is to get a 'library' of
integrals that we know converge or don't. The examples that we did in
class are a good place to start with for this: we did integrals of
1/
x2, 1/
x, and
e-x (all for integrals with
limits going to infinity). The first and last of these converge,
while the middle one doesn't. So the first thing we might remember is
- integrals of 1/xp
from some number (greater than zero) to infinity converge if
p>1.
- integrals of decaying exponentials (e-x, 2-x, etc.) from some number to
infinity converge.
Thus, these functions then become good candidates for comparisons.
Actually figuring out which functions are bigger or smaller than which
of these isn't necessarily easy. A good place to start is by graphing
them or plugging in some points. Also remember that logarithms
(ln(
x), for example) grow amazingly slowly.