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Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx. It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dx= |
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Answer» g(m,N) Put `x=(1)/(1+y)` `rArr""g(m,n)=int_(oo)^(0)(1)/((1+y)^(m-1))(1-(1)/(1+y))^(n-1)(-(1)/((1+y)^(2)))dy` `""=int_(0)^(oo)(y^(n-1))/((1+y)^(m+n))dy` `""=int_(0)^(oo)(x^(n-1))/((1+x)^(m+n))dx` |
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