Evaluate the integral to three decimal places. $$\int _ { e } ^ { 8 } \frac { d x } { x \ln x }$$
A) $0.732$
B) $0.194$
C) $0.094$
D) $0.533$
E) $0.183$

Question

Quizplus · Accepted Answer

The integral $\int \frac{1}{x \ln(x)} dx$ can be solved by recognizing it as the derivative of the logarithmic function. Specifically, if you let $u = \ln(x)$, then $du = \frac{1}{x}dx$, and the integral becomes $\int \frac{1}{u} du$, which is $\ln|u| + C$ or $\ln|\ln(x)| + C$. Evaluating this from $e$ to $8$ gives $\ln|\ln(8)| - \ln|\ln(e)| = \ln(\ln(8)) - \ln(1) = \ln(\ln(8))$. Plugging in the values, $\ln(\ln(8)) \approx 0.732$, making option A correct.

Evaluate the Integral to Three Decimal Places ∫e8dxxln⁡x\int _ { e } ^ { 8 } \frac { d x } { x \ln x }∫e8​xlnxdx​

Evaluate the Integral to Three Decimal Places $\int _ { e } ^ { 8 } \frac { d x } { x \ln x }$