Question 14055. (Q1507211)

\[ \begin{aligned} & \int_0^{\frac 12 \pi } \log \sin x \sqrt {\sin x } \,dx = \frac {(\pi - 4) (2\pi)^3}{\{ \varGamma (\frac 14) \}^2 },\\ & \int_0^{\frac 12 \pi } \frac {\log \sin x dx }{\sqrt {\sin x }} = -2^{- \frac 32} \sqrt \pi\;\{\varGamma (\tfrac 14) \}^2 ,\\ & \int_0^{\frac 12 \pi } \frac {( \log \text{sec\,} \frac 12 + \text{tg\,} \frac 12\, x) dx }{\sqrt {\sin x }} = 2^{\frac 52} \sqrt \pi \{ \varGamma (\tfrac 14)\}^2,\\ & \int_0^{\frac 12 \pi } \sin x (\log \sin x)^2 dx = (\log 2 -1 )^2 + 1 \,\tfrac 1{12}\, \pi^2.\end{aligned} \]