Paragraph: If m 0, n 0 , the definite integral I = ^1 x ^ m -1 (1- x )^ n -1 dx depends upon the values of m and n and is denoted by ( m , n ) , called the beta function. e.g. ₀^1 x^4(1-x)^5 d x= ₀^1 x⁵⁻¹(1-x)⁶⁻¹ d x= (5,6) and ₀^1 x^ 5 / 2

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Paragraph: If m 0, n 0 , the definite integral I = ^1 x ^ m -1 (1- x )^ n -1 dx depends upon the values of m and n and is denoted by ( m , n ) , called the beta function. e.g. ₀^1 x^4(1-x)^5 d x= ₀^1 x⁵⁻¹(1-x)⁶⁻¹ d x= (5,6) and ₀^1 x^ 5 / 2 (1-x)^ -1 / 2 d x= ₀^1 x^ 7 / 2-1 (1-x)^ 1 / 2-1 d x= ( 7 2 , 1 2 ) Obviously, ( n , m )= ( m , n ) . Question: If ₀^n (1- x n )^n x^ k-1 d x=R (k, n+1) , then R is equal to

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Correct answer: C. n ^k

Writing x n =2 , we get ₀^n (1- x n )^n x^ k-1 d x= ₀^1(1-z)^n(n z)^ k-1 n d z=n^k ₀^1(1-z)^n z^ k-1 d z=n^k (k, n+1)

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