$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$
A) $\frac { 1 } { 8 } \ln ( 5 )$
B) $\frac { 8 } { \ln ( 5 ) }$
C) $\frac { - 1 } { 5 } \ln ( 8 )$
D) $8 \ln ( 5 )$
E) $\frac { 1 } { 5 } \ln ( 8 )$

Question

$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$ 
A) $\frac { 1 } { 8 } \ln ( 5 )$
B) $\frac { 8 } { \ln ( 5 ) }$
C) $\frac { - 1 } { 5 } \ln ( 8 )$
D) $8 \ln ( 5 )$
E) $\frac { 1 } { 5 } \ln ( 8 )$

Quizplus · Accepted Answer

This integral can be solved by recognizing it as a form suitable for substitution with $u = \ln(x)$, which gives $du = \frac{1}{x}dx$. The integral becomes $\int_1^5 \frac{1}{u^8} du$, which evaluates to $\frac{1}{7u^7}$ from 1 to 5, simplifying to $\frac{1}{8}\ln(5)$.

Evaluate the definite integral ∫ee51xln⁡(x)8dx. \text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. } Evaluate the definite integral ∫ee5​xln(x)81​dx.

$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$