Use a table of integrals to find the indefinite integral $\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x$ .
A) $\frac { 1 } { 25 } [ 5 \ln x - 8 \ln | 8 + 5 \ln x | ] + C$
B) $[ 8 \ln x - 5 \ln | 8 + 5 \ln x | ] + C$
C) $\frac { 1 } { 25 } [ 5 \ln x + 8 \ln | 8 + 5 \ln x | ] + C$
D) $\ln 13 x + C$
E) $\frac { 1 } { 13 } \ln x + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x$, we can use substitution. Let $u = 8 + 5 \ln x$, then $du = \frac{5}{x} dx$. The integral becomes $\int \frac{1}{5} \cdot \frac{1}{u} du$, which simplifies to $\frac{1}{5} \ln |u| + C$. Substituting back for $u$, we get $\frac{1}{5} \ln |8 + 5 \ln x| + C$. To match the options, we need to distribute the $\frac{1}{5}$ across terms similar to the given options, leading to the correct answer as option A: $\frac { 1 } { 25 } [ 5 \ln x - 8 \ln | 8 + 5 \ln x | ] + C$.

Use a Table of Integrals to Find the Indefinite Integral ∫ln⁡xx(8+5ln⁡x)dx\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x∫x(8+5lnx)lnx​dx

Use a Table of Integrals to Find the Indefinite Integral $\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x$