If m > 0, n > 0, the definite integral I = ∫xm - 1(1 - x)n - 1dx for x ∈ [0, 1] depends upon the values of m and n and is denoted by b (m, n), called the beta function. That is ∫x4(1 - x)5dx for x ∈ [0, 1] = ∫x5 - 1(1 - x)6 - 1dx for x ∈ [0, 1] = β(5, 6) and ∫x5/2(1 - x)-1/2dx for x ∈ [0, 1] = ∫x7/(2 - 1)(1 - x)1/(2 - 1)dx for x ∈ [0, 1] = β(7/2, 1/2). Obviously, β(n, m) = β(m, n).
If ∫(1 - x/n)n xk - 1dx for x ∈ [0, n] = Rβ(k, n + 1), then R is equal to
(A) n
(B) nkn
(C) nk
(D) of these
