Evaluate the integral. The integral may not require integration by parts.
-$$\int \frac { \ln x } { x ^ { 4 } } d x$$
A) $- \frac { 1 } { 3 x ^ { 3 } } \ln x - \frac { 1 } { 9 x ^ { 3 } } + C$
B) $\frac { 5 } { x ^ { 6 } } + C$
C) $- \frac { 1 } { 3 } \ln x - \frac { 1 } { 3 x ^ { 3 } } + C$
D) $- \frac { 1 } { x } \ln x - \frac { 1 } { 9 x ^ { 3 } } + C$

Question

Quizplus · Accepted Answer

This integral can be solved by recognizing it as a case for integration by parts, where $u = \ln x$ and $dv = x^{-4} dx$. However, a more straightforward approach is to use the substitution $u = \ln x$, which simplifies the integral to $\int u x^{-4} dx$. The derivative of $u = \ln x$ is $du = \frac{1}{x} dx$, but directly integrating $\frac{\ln x}{x^4}$ without substitution also leads to the result $-\frac{1}{3x^3}\ln x - \frac{1}{9x^3} + C$, which matches option A. This is because the integral involves a natural logarithm function divided by a power of $x$, which typically results in a logarithmic term and a polynomial term in the answer.

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