Given that for each `a in (0,1) lim^(h to 0^(+)) int_(h)^(1-h) t^(-a)(1-t)^(a-1)dt` exists. If this limit be g(a), then the value `g((1)/(2))`, is

Options

A. `pi`
B. `2pi`
C. `(pi)/(2)`
D. `(pi)/(4)`

Answer

A

Explanation

We have,
`g(a)=overset(ht o 0^(+))lim overset(1-h)underset(h)int t^(-a)(1-t)^(a-1)dt`
`:. G((1)/(2))=overset(h to 0^(+))lim overset(1-h)underset(h)int t^(-1//2)(1-t)^(-1//2)dt`
`rArr g((1)/(2))=underset(h to 0^(+))lim overset(1-h)underset(h)int (1)/(sqrt(t-t^(2))dt`
`rArr g((1)/(2))=underset(h to 0^(+))lim overset(1-h)underset(h)int (1)/(sqrt(((1)/(2))^(2)-(t-(1)/(2))^(2))dt`
`rArr g((1)/(2))=underset(1)underset(0)int (1)/(sqrt(((11)/(2))^(2)-(t-(1)/(2))^(2))`
`rArr g((1)/(2))=[sin^(-1)((t-(1)/(2))/(1//2))]_(0)^(1)=sin^(-1)1-sin^(-1)(-1)=pi`