Prove that `int_(0)^(1) x^m(1-x)^ndx=int_(0)^(1) x^n(1-x)^mdx`

Prove that int_(0)^(1)xe^(x)dx=1

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

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

int_(0)^(1)x(1-x)^(4)dx=(1)/(30)

Prove that int_(-1)^(1)x^(17)cos^(4)xdx=0

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

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

int_(0)^(1)x sin(x-1)dx+int_(0)^(1)(1-x)sin xdx=