$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$
A) $\frac { ( \ln x ) ^ { 7 } } { 6 } + C$
B) $\frac { ( \ln x ) ^ { 7 } } { 7 } + C$
C) $\frac { ( \ln x ) ^ { 6 } } { 6 } + C$
D) $6 ( \ln x ) ^ { 5 } + C$
E) $\frac { 7 \ln x } { x } + C$

Question

$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$ 
A) $\frac { ( \ln x ) ^ { 7 } } { 6 } + C$
B) $\frac { ( \ln x ) ^ { 7 } } { 7 } + C$
C) $\frac { ( \ln x ) ^ { 6 } } { 6 } + C$
D) $6 ( \ln x ) ^ { 5 } + C$
E) $\frac { 7 \ln x } { x } + C$

Quizplus · Accepted Answer

This integral can be solved by recognizing that the derivative of $\ln(x)$ is $\frac{1}{x}$, making $\ln(x)$ a natural choice for a substitution in an integral involving $\frac{(\ln(x))^6}{x}$. Using the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$), we find that the integral of $\frac{(\ln(x))^6}{x}$ dx is $\frac{(\ln(x))^7}{7} + C$, since incrementing the power from 6 to 7 and dividing by the new power gives us the correct antiderivative.

Find the indefinite integral ∫(ln⁡x)6xdx\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x Find the indefinite integral ∫x(lnx)6​dx

$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$