Question #41529

The question tests the application of the Tangent Half-Angle (Weierstrass) substitution combined with the $f(t) + tf'(t)$ integration form under a modulus.

Students often blindly apply the substitution $t = \tan(x/2)$ over $[0, 2\pi]$ without splitting the integral at the discontinuity $x = \pi$, leading to a trivial zero bound $[t^2 e^{-t^2}]_0^0 = 0$.

Furthermore, the modulus forces an analysis of the sign of $t(1-t^2)$, which changes at $t=1$ and $t=-1$ (corresponding to $x=\pi/2$ and $3\pi/2$).

Failing to account for these sign inversions and the infinite bounds at $x=\pi$ invalidates the direct use of the reverse product rule, trapping students who ignore piecewise boundary conditions.