`int_(0)^(1)x^((m-1))(1-x)^((n-1))dx` is equal to where ` m,n in N`

If m, n in N , then l_(m n) = int_(0)^(1) x^(m) (1-x)^(n) dx is equal to

int_(0)^(1)(log(1)/(x))^(n-1)dx equals

If I(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx, then

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . If int_(0)^(n)(1-(x)/(n))^(n)x^(k-1)dx=R beta(k, n+1) , then R is equal to

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . The integral int_(0)^(pi//2)cos^(2m)theta sin^(2n) theta d theta is equal to

If I(m,n)=int_0^1x^(m-1)(1-x)^(n-1)dx , then

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-x)^(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)^(1)(x^(m-1)+x^(n-1))/((1+x)^(m+n))dx=