Let $k$ be a natural number. What are some good ways to integrate $\int_0^1 x^{k+1} (1-x)^k dx$ and $\int_0^1 x^{k+2} (1-x)^k dx$? I wanted to use some symbolic math software but I don't have one, so I have to learn a way to do it manually. Thanks!
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2See here: http://math.stackexchange.com/questions/251624/problem-deriving-beta-distribution-normalizing-constant – Feb 26 '13 at 16:49
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You can try http://wolframalpha.com, maxima is quite capable. SAGE and PARI/GP are integrated ssytems. – vonbrand Feb 26 '13 at 17:39
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For a very complete treatment of explicit integration methods, see the freely available 1921 book A Treatise on the Integral Calculus by Joseph Edwards. – Dave L. Renfro Feb 27 '13 at 20:39
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Let $$I(a,b)=\int_0^1x^a(1-x)^bdx $$ $$I(0,b)=\frac{1}{b+1}\quad I(a,b)=\frac{a}{b+1}I(a-1,b+1)$$ The second identity is by integrating by parts. Solving,$$I(a,b)=\frac{a!}{(b+1)(b+2)...(b+a+1)}=\frac{a!b!}{(a+b+1)!}$$ which solves the problem.
Ishan Banerjee
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4A nice check is that the final answer has to be symmetric in $a$ and $b$ as you can define $u=1-x$ and get the same integral with $a$ and $b$ interchanged. – Ross Millikan Feb 26 '13 at 17:04
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Hints: this is the Beta Function:
$$\int\limits_0^1 x^{k+1}(1-x)^k=B(k+2\,,\,k+1)=\frac{\Gamma(k+2)\Gamma(k+1)}{\Gamma(2k+3)}=\frac{(k+1)!k!}{(2k+2)!}$$
DonAntonio
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