Classical Integral.

freemind

Classical Integral.

**Posted:** Sat Mar 01, 2008 4:35 pm

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didilica · Yang-Mills Theory Offline Joined: 06 Mar 2006 Posts: 685

$E=\frac{\pi}{2008}$ .

We have that

$E=\frac{1}{2}\beta(502, 1/2)\frac{1}{2}\beta(1005/2,1/2)$

$=\frac{1}{4}\frac{\Gamma(502)\Gamma(1/2)}{\Gamma(1005/2)}\frac{\Gamma(1005/2)\Gamma(1/2)}{\Gamma(503)}$

$=\frac{\pi}{4\cdot 502}=\frac{\pi}{2008}.$

**Posted:** Sat Mar 01, 2008 4:56 pm

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Didi

Yuriy Solovyov

#3

$\int{\cos^{n}{x} dx}=\frac{\cos^{n-1}{x}\sin {x}}{n}+\frac{n-1}{n}\int{\cos^{n-2}{x} dx}$
So we have:
$\int_{0}^{\frac{\pi}{2}}{\cos^{n}{x} dx}=\frac{n-1}{n}\int_{0}^{\frac{\pi}{2}}{\cos^{n-2}{x} dx}$
We have:
$\int_{0}^{\frac{\pi}{2}}{\cos^{1003}{x} dx}=\frac{2\cdot 4\cdot 6\cdot ... \cdot 1002}{3\cdot 5\cdot 7\cdot ...\cdot 1003}\in...$
$\int_{0}^{\frac{\pi}{2}}{\cos^{1004}{x} dx}=\frac{3\cdot 5\cdot 7\cdot ... \cdot 1003}{2\cdot 4\cdot 5\cdot ...\cdot 1004}\in...$
We will have:
$E=\frac{2\cdot 4\cdot 6\cdot ...\cdot 1002}{3\cdot 5\cdot 7\cdot ... \cdot 1003}\cdot \frac{3\cdot 5\cdot 7\cdot ... \cdot 1...$

**Posted:** Mon Mar 03, 2008 9:49 pm