Evaluate the integral.
$$\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$$
A) $\frac { 1 } { 2 }$
B) $- \frac { 1 } { 4 }$
C) $\frac { 1 } { 4 }$
D) divergent

Question

Evaluate the integral.
$$\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$$
A) $\frac { 1 } { 2 }$ 
B) $- \frac { 1 } { 4 }$
C) $\frac { 1 } { 4 }$
D) divergent

Quizplus · Accepted Answer

To evaluate this integral, we need to determine if $\frac{1}{x\ln x}$ approaches zero as $x$ approaches infinity. We can use the limit comparison test, comparing the given integral to $\int_1^\infty \frac{1}{x}$:
$$\lim_{x \to \infty} \frac{\frac{1}{x\ln x}}{\frac{1}{x}} = \lim_{x \to \infty} \frac{x}{\ln x} = \infty$$
Since the limit diverges, we cannot use the integral test and the given integral is divergent. Therefore, the best choice is D.

Evaluate the Integral ∫1∞dxxln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }∫1∞​xlnxdx​

Evaluate the Integral $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$