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If m gt 0, n gt 0, the definite integral...

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)`.
If `int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dx`, then k is equal to

A

`(m)/(n)`

B

1

C

`(n)/(m)`

D

none of these

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The correct Answer is:
B
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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) . If int_(0)^(n)(1-(x)/(n))^(n)x^(k-1)dx=R beta(k, n+1) , then R is equal to

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If mgt0, ngt0 , the definite integral I=int_0^1 x^(m-1)(1-x)^(n-1)dx depends upon the values of m and n is denoted by beta(m,n) , called the beta function.For example, int_0^1 x^3 (1-x)^4dx=int_0^1 x^(4-1) (1-x)^(5-1) dx=beta(4,5) and int_0^1 x^(3/2) (1-x)^((-1)/2)dx=int_0^1 x^(5/2-1) (1-x)^(1/2-1)dx=beta(5/2,1/2) .Obviously, beta(n,m)=beta(m,n) .Now answer the question:If int_0^oo x^(m-1)/(1+x)^(m+n)dx=k int_0^oo x^(n-1)/(1+x)^(m+n)dx , then k is equal to (A) m/n (B) 1 (C) n/m (D) none of these

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

If mgt0, ngt0 , the definite integral I=int_0^1 x^(m-1)(1-x)^(n-1)dx depends upon the values of m and n is denoted by beta(m,n) , called the beta function.For example, int_0^1 x^3 (1-x)^4dx=int_0^1 x^(4-1) (1-x)^(5-1) dx=beta(4,5) and int_0^1 x^(3/2) (1-x)^((-1)/2)dx=int_0^1 x^(5/2-1) (1-x)^(1/2-1)dx=beta(5/2,1/2) .Obviously, beta(n,m)=beta(m,n) .Now answer the question:If int_0^2 (8-x^3)^((-1)/3)dx=kbeta(1/3,2/3) , then k equals to (A) 1 (B) 1/2 (C) 1/3 (D) 1/4

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FIITJEE-TIPS-PARAGRAPH BASED (MULTIPLE CHOICE) (COMPREHENSION - I)
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