Passage

Given that for each $$\displaystyle a  \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$$ exists. Let this limit be $$g(a)$$ 
In addition, it is given that the function $$g(a)$$ is differentiable on $$(0, 1)$$
Then answer the following question.
Mathematics

The value of $$g\displaystyle \left ( \frac{1}{2} \right )$$ is?


ANSWER

$$\pi$$


SOLUTION
$$g \displaystyle \left ( \frac{1}{2} \right ) = \lim_{h \rightarrow 0^+} \int_h^{1 - h} t^{-1/2} (1-t)^{-1/2} dt$$
$$=

\int_0^1 \displaystyle \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 ( \frac{t - \frac{1}{2}}{\frac{1}{2}} \right ) |_0^1$$
$$= sin^{-1} 1 -sin^{-1} (-1) = \pi$$
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Single Correct Hard Published on 17th 09, 2020
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