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Question Number 159217 by mathlove last updated on 14/Nov/21

Answered by mnjuly1970 last updated on 15/Nov/21

$$solution..$$$$\text{Missing left or extra right}$$$$=2{lim}_{x\to \frac{\pi }{6}}\frac{cos\left(x\right){\mathrm{\Gamma }}^{\prime }\left(sin\left(x\right)\right)}{\mathrm{\Gamma }\left(sin\left(x\right)\right)}.\frac{{sin}^2\left(x\right)}{(-cos\left(x\right)){\mathrm{\Gamma }}^{\prime }\left(\frac{1}{sin\left(x\right)}\right)}$$$$=-2\{\frac{\frac{\sqrt{3}}{2}.\mathrm{\Gamma }\left(\frac{1}{2}\right)\psi \left(\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\right)}.\frac{\frac{1}{4}}{\frac{\sqrt{3}}{2}\left(\mathrm{\Gamma }\left(2\right)\psi \left(2\right)\right)}\}$$$$=\frac{1}{2}.\frac{(-\gamma -2ln\left(2\right))}{1-\gamma }$$

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