Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
-$\int _ { 2 } ^ { \infty } \frac { d x } { x ( \ln x ) ^ { 2 } }$
A) 2 ln 2
B) divergent
C) $\frac { 1 } { \ln 2 }$
D) $\frac { 1 } { 2 }$ $( \ln 2 ) ^ { 2 }$

Question

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { 2 } ^ { \infty } \frac { d x } { x ( \ln x ) ^ { 2 } }$
A) 2 ln 2
B) divergent
C) $\frac { 1 } { \ln 2 }$
D) $\frac { 1 } { 2 }$ $( \ln 2 ) ^ { 2 }$

Quizplus · Accepted Answer

To evaluate the given improper integral $\int _ { 2 } ^ { \infty } \frac { d x } { x ( \ln x ) ^ { 2 } }$, we can use substitution. Let $u = \ln x$, which implies $du = \frac{1}{x} dx$. The limits of integration change accordingly: when $x = 2$, $u = \ln 2$, and as $x$ approaches infinity, $u$ approaches infinity. Thus, the integral becomes $\int _ { \ln 2 } ^ { \infty } \frac { d u } { u ^ { 2 } }$. This is a standard integral that evaluates to $-\frac{1}{u}$ from $\ln 2$ to $\infty$, resulting in $\frac{1}{\ln 2}$.

Evaluate the Improper Integral Whenever It Is Convergent ∫2∞dxx(ln⁡x)2\int _ { 2 } ^ { \infty } \frac { d x } { x ( \ln x ) ^ { 2 } }∫2∞​x(lnx)2dx​

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