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Given that for each `a epsilon(0,1),lim_(hto 0^(+)) int_(h)^(1-h)f^(-a)(1-t)^(a-1)dt` exists. Let this limit be `g(a)`. In addition it is given the function `g(a)` is differentiable on`(0,1)`.
The value of `g(1/2)` is

A

`pi`

B

`2pi`

C

`(pi)/2`

D

`(pi)/4`

Text Solution

Verified by Experts

The correct Answer is:
A
`g(1/2)=lim_(kto 0^(+))int_(k)^(1+k)t^(-1//2)(1-t)^(-1//2)dt`
`=int_(0)^(1)(dt)/(sqrt(t-t^(2)))=int_(e)^(1)(dt)/(sqrt(1/4-(t-1/2)^(2)))=sin^(-1)((t-1/2)/(1/2))|._(0)^(1)`
`=sin^(-1)1-sin^(-1)(-1)=pi`
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