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

Quick Computation:

• $I_2$ = \frac{1}{2}$$\frac{pi}{2} = pi/4
• $I_3$ = $\frac{2}{3}$(1) = 2/3
• $I_4$ = \frac{3}{4}$$\frac{pi}{4} = 3pi/16
• $I_5$ = \frac{4}{5}$$\frac{2}{3} = 8/15
• $I_6$ = \frac{5}{6}$$\frac{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$\frac{0 to pi}{2}$ $sin^m$ x $cos^n$ x dx: use Wallis with combined formula

Beta Function Shortcut: integral$\frac{0 to pi}{2}$ sin^(2a-1) x cos^(2b-1) x dx = Gamma(a)*Gamma$\frac{b}{(2*Gamma(a+b)}$) For integer m, n: integral$\frac{0 to pi}{2}$ $sin^m$ 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)