how to use the comparison test to find p

Question

Use the Comparison Test to determine for what values of p the integral 2 to infinity of 1 all over x raised to p times the ln(x) dx converges.

Answer 1

∫₂^∞1/x^plnxdx = lim a→∞∫₂^a1/x^plnxdx = lim a →∞ [(x^-p+1/(-p+1)lnx₂^a) - ∫₂^a(x^-p+1/(- p+1)·1/xdx\] =

lim a→∞[a^-p+1/(-p + 1)lna - 2^-p+1/(-p +1)ln2 - (x^-p+1)/(-p+1)²₂^a)] = lim a→∞[a^{-p + 1}/(-p+1)lnp -2^{-p +1}/(-p+1)ln2 -

a^-p+1/(-p+1)² + 2^-p+1/(-p+1)²]. We see that this limit exis if - p + 1 < 0, p > 1 and integral converges to

∫₂^∞1/x^plnxdx = 2^-p+1/(-p+1)(1/(-p + 1) - ln2)