Does the following integrals converge, and what test should I apply?
For the first change the variable to $\displaystyle u=1/x$ then it should be relativly easy, and what method you use depends on what you have been taught, I would look at the asymtotic behaviiour at zero and infinity.
For one thing it gets rid of the infinite number of oscillations of the integrand at the lower limit of integration, leaving you with a more conventional problem.