Reduction for sin^n x on [0, pi/2]: I_n = ((n-1)/n) * I_(n-2) I_0 = pi/2, I_1 = 1

Reduction for integral(0 to pi/2) sin^m x cos^n x dx: = [(m-1)(m-3)...(n-1)(n-3)...] / [(m+n)(m+n-2)...] * K where K = pi/2 if both m,n are even, K = 1 otherwise.

integral(0 to pi) sin^n x dx = 2*I_n (for all n, since sin^n x is symmetric about pi/2 via King's Rule)

integral(0 to pi/2) tan^n x dx = integral(0 to pi/2) cot^n x dx (by King's Rule with pi/2)