Difference between revisions of "Wallis's formula"
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'''Wallis's formula''' states that | '''Wallis's formula''' states that | ||
| − | (1)<math>\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{ | + | (1)<math>\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)</math> for even <math>n\geq2</math> |
| − | (2)<math>\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{ | + | (2)<math>\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\cdots\left(\frac{n-1}{n}\right)</math> for odd <math>n\geq3</math> |
| − | </math> for odd n | ||
| − | Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated definite integrals | + | Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated definite integrals to more manageable ones. |
Revision as of 04:10, 9 July 2006
Wallis's formula states that
(1)
for even 
(2)
for odd 
Wallis's formula often works well in combination with trigonometric substitution in reducing complicated definite integrals to more manageable ones.
