Given that for each `a epsilon(0,1),lim(hto 0^(+)) int_(h)^(1-h)t^(-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)/2` b. `pi` c.`-(pi)/2` d. `0`

Options

A. a. `pi`
B. b. `2pi`
C. c. `(pi)/2`
D. d. `(pi)/4`

Answer

A

Explanation

`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`