Question #41199

This question tests the student's ability to correctly transition from an implicit algebraic complex number to its explicit trigonometric form using De Moivre's Theorem, while rigorously handling the principal argument and branch cuts.

The core trap lies in the evaluation of the infinite sine series (derived via the Taylor series of the complex logarithm).

Students frequently memorize the standard Fourier series result $\sum \frac{\sin(n\theta)}{n} = \frac{\pi - \theta}{2}$ without the domain restriction $\theta \in (0, 2\pi)$.

For negative $x$, the argument $\theta = 2\tan^{-1} x$ falls in $(-\pi, 0)$.

Because the modulus of the complex logarithm must be positive, a phase shift of $\pi$ occurs, requiring the odd extension of the series which yields $-\frac{\pi + \theta}{2}$.

Failing to account for this branch cut leads to a sign error in the explicit function $g(x)$, resulting in the trap answer $64$ instead of $-64$ after applying the Chain Rule.