Wallis' Formula and Trigonometric Integrals

Wallis' Formula: I_n = integral(0 to pi/2) sin^n x dx = integral(0 to pi/2) cos^n x dx I_n = ((n-1)/n) * I_(n-2) with I_0 = pi/2, I_1 = 1

Quick Computation:

I_2 = (1/2)(pi/2) = pi/4
I_3 = (2/3)(1) = 2/3
I_4 = (3/4)(pi/4) = 3pi/16
I_5 = (4/5)(2/3) = 8/15
I_6 = (5/6)(3pi/16) = 5pi/32

Rule: Even n ends with pi/2 factor. Odd n ends with 1.

Extended Results:

integral(0 to pi) sin^n x dx = 2*I_n (for all n)
integral(0 to 2pi) sin^n x dx = 4*I_n (n even), 0 (n odd)
integral(0 to pi/2) sin^m x cos^n x dx: use Wallis with combined formula

Beta Function Shortcut: integral(0 to pi/2) sin^(2a-1) x cos^(2b-1) x dx = Gamma(a)Gamma(b)/(2Gamma(a+b)) For integer m, n: integral(0 to pi/2) sin^m x cos^n x dx = [(m-1)!!(n-1)!!]/[(m+n)!!] * K where K = pi/2 if both m,n even; K = 1 otherwise. (!! denotes double factorial)