Question #41460

This problem tests the ability to recognize hidden trigonometric ratios within logarithmic integrals.

Students who attempt direct integration by parts on $\ln|1+2\cos 2x|$ will encounter a divergent integral $\int \frac{1}{1+2\cos t} dt$ that requires Cauchy Principal Value to evaluate, often leading to incorrect conclusions or abandoned attempts.

The intended, elegant path requires identifying $1+2\cos 2x = \frac{\sin 3x}{\sin x}$ to split the logarithm, followed by exploiting the periodicity and symmetry of $\ln|\sin t|$ to annihilate the $3x$ term via a three-part interval split.