Demonstrate that the integral of \begin{equation} \cos (y) \left(\sqrt{4-\sin ^2(y)} \cos ^{-1}(\sin (y))+4 \cos (y) \csc ^{-1}(2 \csc (y))\right) \end{equation} over $y \in [0,\frac{\pi}{2}]$ equals \begin{equation} -\text{Li}_2\left(\frac{1}{4}\right)-\frac{3}{2}+\frac{\pi ^2}{3}-2 \log ^2(2)+\frac{15 \log (3)}{8} \approx 2.621207508030023088, \end{equation} where the polylogarithm function is employed. The origin of this assertion (in which I have confidence, but have yet been unable to formally demonstrate) is somewhat of a "long story", but I can expand upon its details.
2026-03-28 01:46:18.1774662378
Evaluate a certain one-dimensional integral involving inverse trigonometric functions
47 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in INTEGRATION
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