Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration
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: The integral ₀^ / 2 ^ 2 m ^ 2 n ~d is equal to
Options
- A1 2 (m+ 1 2 , n+ 1 2 )
- B2 (2 ~m , 2 n )
- C(2 m+1,2 n+1)
- Dnone of these
Correct answer
A. 1 2 (m+ 1 2 , n+ 1 2 )
Step-by-step solution
Writing ^2 = x . we get 2 ~d = dx , and hence the given integral is equal to 1 2 ₀^ / 2 ^ 2 m-1 ^ 2 n-1 (2 ) d = 1 2 ₀^1 ( ^2 )^ 2 m-1 2 ( ^2 )^ 2 n-1 2 dx = 1 2 ₀^1(1-x)^ m-1 / 2 x^ n-1 / 2 d x= 1 2 (m+ 1 2 , n+ 1 2 )