Question

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)

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