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

Question

Aicharya · Accepted Answer

g 1 2 = lim h &#8594; 0 + &#8289; &#8747; h 1 - h t - 1 2 &#160; 1 - t - 1 2 &#160; d t = &#8747; 0 1 d t t - t 2 = &#8747; 0 1 d t 1 4 - t - 1 2 2 = sin - 1 &#8289; t - 1 2 1 2 0 1 = sin - 1 &#8289; 1 - sin - 1 &#8289; (-1) = &#960;

Paragraph: Given that for each $a \in(0,1)$, $\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d…

Solution