If `int_(0)^(1)(x^(21)+x^(14)+x^(7))(2x^(14)+3x^(7)+6)^(1/7)dx=(1)/(l)(11)^(m/n)` where `l,m,n in N,m` and `n` are co-prime then `l+m+n` is equal to

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

int(x^(7m)+x^(2m)+x^(m))(2x^(6m)+7x^(m)+14)^((1)/(m))dx

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

If f(x)=((x^(l))/(x^(m)))^(l+m)((x^(m))/(x^(n)))^(m+n)((x^(n))/(x^(l)))^(n+l) then f'(x)

((x^(m))/(x^(n)))^(m+n)*((x^(n))/(x^(l)))^(n+l)*((x^(l))/(x^(m)))^(l+m)

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 a+b=1,a^(2)+b^(2)=2 and given that a^(7)+b^7=(m)/(n) where m & n are co-primes. Find m+n

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