Question #40449

This question tests the rigorous application of the Leibniz Integral Rule and domain analysis.

A common trap is blindly writing $\sqrt{1-\sin^2 x} = \cos x$ and canceling it, yielding $2\sin^2 x = 1$ (sum of roots $= 4\pi$).

An even more dangerous trap occurs if a student sets the unsimplified derivative to zero and squares the equation $2\sin^2 x \cos x = \sqrt{1-\sin^2 x}$ to get $\cos^2 x(4\sin^4 x - 1) = 0$.

This algebraic manipulation introduces extraneous roots ($\pi/2, 3\pi/2, 3\pi/4, 5\pi/4$) that are either points of non-differentiability or invalid under the sign of $\cos x$, leading to the trap answer $6\pi$.

Recognizing that $\sqrt{x^2} = |x|$ and checking differentiability at the trigonometric boundaries is crucial to identifying the true stationary points.