Answer:
Option A
Explanation:
Plan $\int_{}^{} \frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\left(\frac{x}{a}\right)+c$
As g(a) is defined in the question, first use the numerical value of 'a' given in the question and then proved.
Given, g(a) $=\lim_{h \rightarrow 0}\int_{h}^{1-h}t^{-a} (1-t)^{a-1}dt$
$\therefore$ g(1/2)
$=\lim_{h \rightarrow 0+}\int_{h}^{1-h} t^{-1/2}(1-t)^{-1/2}dt$
$= \int_{0}^{1} \frac{dt}{\sqrt{t-t^{2}}}=\int_{0}^{1} \frac{dt}{\sqrt{\frac{1}{4}-\left(t-\frac{1}{2}\right)^{2}}}$
= $\sin^{-1}\left[\left(\frac{t-1/2}{1/2}\right)\right]^{1}_{0}$
= $\sin^{-1}1-\sin^{-1}(-1)=\pi$
