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{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)</math> for even <math>n\geq2</math> | + | (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 integer]]s <math>n\geq2</math> |
| − | (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> | + | (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 integer]]s <math>n\geq3</math> |
| − | Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated definite | + | Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated [[definite integral]]s to more manageable ones. |
Latest revision as of 17:03, 12 October 2006
Wallis's formula states that
(1)
for even integers 
(2)
for odd integers 
Wallis's formula often works well in combination with trigonometric substitution in reducing complicated definite integrals to more manageable ones.
