Determine if the integral

converges or diverges.

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

also diverges.

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.