Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { \ln | x | } { x } d x$$
A) Converges
B) Diverges

Question

Quizplus · Accepted Answer

We can test for convergence using the integral test:

\begin{align*}
\int _ { 1 } ^ { \infty } \frac { \ln | x | } { x } d x &= \lim_{b 	o \infty} \int _ { 1 } ^ { b } \frac { \ln | x | } { x } d x \
&= \lim_{b 	o \infty} [\ln |x| \cdot \ln |x|]_1^b\
&= \lim_{b 	o \infty} (\ln |b| \cdot \ln |b| - \ln 1 \cdot \ln 1)\
&= \infty
\end{align*}

Since the integral diverges, the answer is B, diverges.

Determine Whether the Improper Integral Converges or Diverges ∫1∞ln⁡∣x∣xdx\int _ { 1 } ^ { \infty } \frac { \ln | x | } { x } d x∫1∞​xln∣x∣​dx

Determine Whether the Improper Integral Converges or Diverges $\int _ { 1 } ^ { \infty } \frac { \ln | x | } { x } d x$