Question #41379

This question tests the student's ability to solve a linear differential equation where the integrating factor creates an apparent singularity at $x=0$.

The primary trap lies in the constant of integration.

Because no explicit initial value is given, students often arbitrarily assume $C=0$ or ignore the constant entirely, leading to a discontinuous function.

By enforcing the condition that $f(x)$ is continuous on the entire interval, the student is forced to apply trigonometric standard limits (L'Hopital's rule logic) to resolve the removable discontinuity, yielding $C=-1$.

A secondary trap is assuming the global extrema occur at the interval boundaries, which yields the distractor $\frac{4\pi}{3} - 2\sqrt{3}$, whereas the true extrema lie at the interior critical points $x = \pm \frac{\pi}{2}$.