Answer
Divergent
Work Step by Step
\[\begin{align}
& \int_{1}^{\infty }{\ln \left| x \right|}dx \\
& \text{From the definition of absolute value} \\
& \int_{1}^{\infty }{\ln \left| x \right|}dx=\int_{1}^{\infty }{\ln x}dx \\
& \text{Then,} \\
& \int_{1}^{\infty }{\ln x}dx=\underset{b\to \infty }{\mathop{\lim }}\,\int_{1}^{b}{\ln x}dx \\
& \text{Integrating} \\
& \text{=}\underset{b\to \infty }{\mathop{\lim }}\,\left[ x\ln x-x \right]_{1}^{b} \\
& \text{=}\underset{b\to \infty }{\mathop{\lim }}\,\left[ \left( b\ln b-b \right)-\left( 1\ln \left( 1 \right)-1 \right) \right] \\
& \text{=}\underset{b\to \infty }{\mathop{\lim }}\,\left[ \left( b\ln b-b \right)+1 \right] \\
& \text{=}\underset{b\to \infty }{\mathop{\lim }}\,\left[ b\left( \ln b-1 \right)+1 \right] \\
& \text{Evaluate when }b\to \infty \\
& \text{=}\infty \cdot \infty+1 \\
& =\infty \\
& \text{ The improper integral diverges} \\
& \text{Divergent} \\
\end{align}\]
