Question #41524

This question tests the integration of logarithmic trigonometric functions combined with definite integral properties.

The primary trap is the modulus inside the logarithm.

Students often ignore the modulus, leading to domain errors since $\sqrt{3}\sin x - \cos x$ is negative on $[0, \pi/6)$.

The non-differentiable extremum at $x = \pi/6$ forces the use of the modulus.

The elegant solution bypasses a messy split-interval integration by recognizing that after a phase shift, the integrand becomes periodic with period $\pi$.

This allows shifting the integration bounds to $[0, \pi]$, where the sine function is positive, naturally resolving the modulus and setting up a clean application of King's Property and integration by parts.