#### 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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

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