Mathematics

Problem

Determine whether the improper integral

\displaystyle \int_{2}^{\infty} \frac{5}{x\, (\ln x)^{2}}\,dx

converges or diverges. a) It converges to

0

. b) It diverges because

\displaystyle \int_{2}^{\infty}\frac{1}{x\ln x}\,dx

diverges, and the given integrand is comparable to that integral. c) It converges to

\dfrac{5}{\ln 2}

. d) It diverges because the integrand is asymptotically equivalent to

\tfrac{1}{x}