Question #41446

This question tests the student's ability to correctly apply reduction formulas while strictly observing boundary conditions, specifically targeting the 'Arbitrary Constants vs Parameters' trap.

Students often memorize Wallis' formula as a parameter-driven sequence and blindly apply it to integrals of the form $\int \cos^n x \, dx$.

Here, the binomial expansion condenses the integrand to $\cos^6(2x)$, and the substitution $2x=t$ shifts the upper limit to $\pi/4$, not $\pi/2$.

The trap lies in assuming the boundary term $\frac{1}{n}[\cos^{n-1}t \sin t]$ vanishes (as it does for $\pi/2$), leading students to treat the recursive parameter as an independent constant and erroneously select Option D (or B, if they miss the $dt/2$ differential factor).

The correct execution requires retaining the non-zero boundary constants at each recursive step.