When we look at this function, we see that it's badly behaved for two reasons. The function is badly behaved at x = 0, and the upper limit of integration is also badly behaved. We split the integral somewhere nice, like x = 1:
Each of these integrals is improper for only one reason. The integral
is improper because of the function being badly behaved, and the integral
is improper because of its upper limit of integration.
Since we know
diverges, we know the original integral
If you look at the formula and the graph of the function you can find all the trouble spots. You've practiced integrating plenty of improper integrals, so these are mostly to practice breaking integrals up.