Calculus Improper Integral ....Converging/Diverging?

It's divergent. Since the antiderivative of x^(−a) is −x^(−(a−1))/(a−1), which, if a > 1, approaches −∞ as x approaches 0, the integral of x^(−a) will diverge at 0 whenever a > 1. (It also diverges if a = 1 as the antiderivative of 1/x is ln x, which also approaches −∞ as x approaches 0.)