Paragraph: Given that for each a (0,1) , _ h 0⁺ _ h ^ 1-h t^ -a (1-t)^ a-1 d t exists. Let this limit be g(a) . In addition, it is given that the function g(a) is differentiable on (0,1) . Question: The value of g ( 1 2 ) is

Step-by-step solution

g 1 2 = lim h → 0 + ⁡ ∫ h 1 - h t - 1 2   1 - t - 1 2   d t = ∫ 0 1 d t t - t 2 = ∫ 0 1 d t 1 4 - t - 1 2 2 = sin - 1 ⁡ t - 1 2 1 2 0 1 = sin - 1 ⁡ 1 - sin - 1 ⁡ (-1) = π