Wallis' formula: integral(0 to pi/2) sin^n(x) dx = integral(0 to pi/2) cos^n(x) dx = W_n

For even n = 2m: W_(2m) = [(2m-1)!!/(2m)!!] * pi/2

For odd n = 2m+1: W_(2m+1) = (2m)!!/(2m+1)!!

Quick values: W_0=pi/2, W_1=1, W_2=pi/4, W_3=2/3, W_4=3pi/16, W_5=8/15, W_6=5pi/32

Extended to [0, pi] and [0, 2pi]:

• integral(0 to pi) sin^n dx = 2*W_n (for all n)
• integral(0 to 2pi) sin^n dx = 4*W_n (n even), 0 (n odd)

Mixed powers: integral(0 to pi/2) sin^m(x)cos^n(x) dx = B((m+1)/2, (n+1)/2)/2

Wallis product: pi/2 = lim [(224466...)/(133557...)]

Common exam application: Direct computation using the chain of ratios. Remember: pi/2 appears only for even n (or when both m,n are even in mixed products).