सिद्ध कीजिए कि
`int_(0)^(1)x^(m)(1-x)^(n)dx=int_(0)^(1)x^(n)(1-x)^(m)dx`

int_(0)^(1) x (1 -x)^(n) dx=?

int_(0)^(1)(1-x^(3))^(n)dx=

evaluate int_(0)^(1)x^(2)(1-x)^(n)dx

U_(n)=int_(0)^(1)x^(n)(2-x)^(n)dx and V_(n)=int_(0)^(1)x^(n)(1-x)^(n)dx,n in N and if (V_(n))/(U_(n))=1024, then the value of n is

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

Evaluate : (i) int_(0)^(1)x(1-x)^(n)dx (ii) int_(0)^(1)x(1-x)^(3//2)dx