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Question Number 95378 by Fikret last updated on 24/May/20

$$\mathrm{If}I(m,n)=\underset{0}{\stackrel{1}{\int }}{x}^{m-1}{(1-x)}^{n-1}dx,\mathrm{then}$$

Commented by Tony Lin last updated on 24/May/20

$$thenI(m,n)isaBetafunction$$

Answered by mathmax by abdo last updated on 25/May/20

$$\Rightarrow \mathrm{I}(\mathrm{m},\mathrm{n})=\mathrm{B}(\mathrm{m},\mathrm{n})=\frac{\mathrm{\Gamma }\left(\mathrm{m}\right).\mathrm{\Gamma }\left(\mathrm{n}\right)}{\mathrm{\Gamma }(\mathrm{m}+\mathrm{n})}$$$$\mathrm{\Gamma }\left(\mathrm{x}\right)={\int }_0^{\mathrm{\infty }}{\mathrm{t}}^{\mathrm{x}-1}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}(\mathrm{x}>0)$$

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