Determine whether the improper integral converges or diverges.

MATHEMATICS · Single correct · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

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}