Question

Given that for each $a \in(0,1), \displaystyle\lim _{h \rightarrow 0^{+}} \int\limits_{h}^{1-h} t^{-a}(1-t)^{a-1} d t$ exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
The value of $g\left(\frac{1}{2}\right)$ is

• A. $\pi$
• B. $2 \pi$
• C. $\frac{\pi}{2}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (A)

$g\left(\frac{1}{2}\right)=\displaystyle\lim _{h \rightarrow 0^{+}} \int\limits_{h}^{1- h } t ^{-1 / 2}(1-t)^{-1 / 2} dt$
$=\int\limits_{0}^{1} \frac{ dt }{\sqrt{ t - t ^{2}}}=\int\limits_{0}^{1} \frac{ dt }{\sqrt{\frac{1}{4}-\left( t -\frac{1}{2}\right)^{2}}} $
$=\left.\sin ^{-1}\left(\frac{ t -\frac{1}{2}}{\frac{1}{2}}\right)\right|_{0} ^{1}$
$=\sin ^{-1} 1-\sin ^{-1}(-1)=\pi$