Question #40405

This question tests the 'Lethal Collision' between variable substitution and the Principal Value Branch of inverse trigonometric functions.

Students often blindly apply the standard identity $\cos^{-1}(4x^3 - 3x) = 3\cos^{-1}x$, which is only valid for $x \in [1/2, 1]$.

For the interval $x \in [-1, -1/2]$, the substituted angle $3\theta$ falls in the range $[2\pi, 3\pi]$, requiring a mandatory quadrant adjustment to $3\theta - 2\pi$.

Failing to account for this specific $2\pi$ phase shift leads directly to an unadjusted integrand of $3\pi$, trapping the student into selecting the distractor $3\pi/2$.