Determine whether the improper integral diverges or converges.Evaluate the integral if it converges.

Question

Determine whether the improper integral  diverges or converges.Evaluate the integral if it converges. ​

Quizplus · Accepted Answer

The integral $\int_{1}^{\infty} \frac{1}{x \ln x}$ can be evaluated using the substitution $u = \ln x$, which gives $\int_{0}^{\infty} \frac{1}{u} du$. This integral diverges, so the original integral also diverges by the comparison test with the p-series $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$. Therefore, the best choice is B, which correctly states that the integral diverges.

Determine Whether the Improper Integral Diverges or Converges