Paragraph: Given that for each $a \in(0,1)$, $\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t$ 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 $g\left(\frac{1}{2}\right)$ is

• Correct answer
  A

  $\pi$
• B

  $2\pi$
• C

  $\frac{\pi }{2}$
• D

  $\frac{\pi }{4}$

Hence, the correct answer is option: 'A'.