Question #41506

This question tests the integration of completing the square with the geometrical interpretation of modulus, while setting two rigorous traps.

First, students often forget that $\sqrt{x^2-4x+4} = |x-2|$, leading them to integrate an odd function over a symmetric interval, resulting in $0$ (Option D).

Second, those who correctly use the substitution $x-2 = \tan \alpha$ often blindly assume $\arcsin(\sin 2\alpha) = 2\alpha$ over the entire interval.

Because the upper limit $\arctan 2$ exceeds $\pi/4$, $2\alpha$ crosses $\pi/2$, moving outside the principal domain of arcsine.

Failing to split the integral and apply the correct branch $\pi - 2\alpha$ leads directly to the extraneous solution trap (Option B).