Wallis' formula: integral$\frac{0 to pi}{2}$ $sin^n$(x) dx = integral$\frac{0 to pi}{2}$ $cos^n$(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$\frac{0 to pi}{2}$ $sin^m$(x)$cos^n$(x) dx = B$\frac{(m+1}{2}$, $\frac{n+1}{2}$)/2

Wallis product: pi/2 = lim [$\frac{2*2*4*4*6*6*...}{(1*3*3*5*5*7*...)}$]

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).