Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
-$\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$
A) divergent
B) $\frac { 2 } { \ln 2 }$
C) 2 ln 2
D) ln 2
E) $\frac { \ln 2 } { 2 }$

Question

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$
A) divergent
B) $\frac { 2 } { \ln 2 }$
C) 2 ln 2
D) ln 2
E) $\frac { \ln 2 } { 2 }$

Quizplus · Accepted Answer

The integral $\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$ can be evaluated using the substitution $u = \ln(x)$, which transforms the integral into $\int \frac{du}{u}$. This integral evaluates to $\ln|u|$ or $\ln|\ln(x)|$, which diverges as $x$ approaches infinity.

Evaluate the Improper Integral Whenever It Is Convergent ∫2∞dxxln⁡x\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }∫2∞​xlnxdx​

Evaluate the Improper Integral Whenever It Is Convergent $\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$