Evaluate the integral. $$\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$$
A) $- \frac { 1 } { 5 } ( \ln 2 ) ^ { 2 }$
B) $- \frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$
C) 0
D) $\frac { 1 } { 6 } ( \ln 3 ) ^ { 2 }$
E) $\frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

The integral's limits of integration are dependent on $x$, but $x$ is also the variable of integration. This means the integral evaluates over a range that changes with $x$, making it impossible to directly evaluate without a specific value for $x$. However, since the question does not provide a specific value for $x$, and given the nature of the integral, it suggests a misunderstanding in the formulation of the problem. In typical integral evaluations, the limits of integration are constants or the variable of integration is different from the variable in the limits. Therefore, without additional context or correction, the integral as presented does not have a meaningful evaluation, leading to the conclusion that none of the provided options (A, B, D, E) correctly represents the value of the integral, suggesting the answer is C) 0, under the assumption that the question intended to present a scenario where the integral evaluates to a null or non-effect (e.g., an integral over a zero-width interval).

Evaluate the Integral ∫x/6x/33ln⁡(tan⁡x)7sin⁡xcos⁡xdx\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x∫x/6x/3​7sinxcosx3ln(tanx)​dx

Evaluate the Integral $\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$