Question

Given that for each $a\in (0,1),\underset{h\to 0^{+}}{\lim }\underset{h}{\overset{1-h}{\int }}{t}^{-a}(1-t{)}^{a-1}dt$ 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)

$g\left(\frac{1}{2}\right)=\underset{h\to 0^{+}}{\lim }\underset{h}{\overset{1-h}{\int }}{t}^{-1/2}(1-t{)}^{-1/2}dt$
$=\underset{0}{\overset{1}{\int }}\frac{dt}{\sqrt{t-t^2}}=\underset{0}{\overset{1}{\int }}\frac{dt}{\sqrt{\frac{1}{4}-{\left(t-\frac{1}{2}\right)}^2}}$
$={{\sin }^{-1}\left(\frac{t-\frac{1}{2}}{\frac{1}{2}}\right)|}_0^1$
$={\sin }^{-1}1-{\sin }^{-1}(-1)=\pi$