The integral ∫cos^2mθsin^2nθdθ for θ ∈ [0, π/2] is equal to is equal to

Question

If m > 0, n > 0, the definite integral I = ∫x^{m - 1}(1 - x)^{n - 1}dx 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] = ∫x^{5 - 1}(1 - x)^{6 - 1}dx for x ∈ [0, 1]  = β(5, 6) and ∫x⁵/2(1 - x)^-1/2dx for x ∈ [0, 1] = ∫x^{7/(2 - 1)}(1 - x)^{1/(2 - 1)}dx for x ∈ [0, 1] = β(7/2, 1/2). Obviously, β(n, m) = β(m, n).

The integral ∫cos^2mθsin²ⁿθdθ for θ ∈ [0, π/2] is equal to  is equal to

(A) 1/2β(m + 1/2, n + 1/2)

(B) 2β(2m, 2n)

(C) β(2m + 1, 2n + 1)

(D) None of these 

Answer 1

Answer is (A) 1/2β(m + 1/2, n + 1/2)

Writing sin²θ = x, we get 2sinθcosθdθ = dx, and hence the given integral is