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Math Help - beta function

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    beta function

    prove that \int_{0}^{1}\displaystyle\frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}}dx=B(m,n)
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  2. #2
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    Quote Originally Posted by Ajreb View Post
    prove that \int_{0}^{1}\displaystyle\frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}}dx=B(m,n)
    first note that the substitution t \rightarrow 1-t gives us: \int_{\frac{1}{2}}^1 t^{m-1}(1-t)^{n-1} \ dt = \int_0^{\frac{1}{2}} t^{n-1} (1-t)^{m-1} \ dt. therefore we have:

    B(m,n)=\int_0^1 t^{m-1}(1-t)^{n-1} \ dt = \int_0^{\frac{1}{2}}[ t^{m-1}(1-t)^{n-1} + t^{n-1}(1-t)^{m-1}] \ dt. now do the substitution: t = \frac{x}{x+1}. \ \Box
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