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Question Number 152201 by mnjuly1970 last updated on 26/Aug/21

$$$$$$...\mathrm{Integral}...$$$$\mathrm{I}:={\int }_0^{\pi }ln\left(sin\left(x\right)\right).{tan}^{-1}\left(cot\left(x\right)\right)dx\stackrel{?}{=}0$$$$proof::....$$$$\mathrm{I}:={\int }_0^{\pi }ln\left(sin\left(x\right)\right).{tan}^{-1}\left(tan(\frac{\pi }{2}-x)\right)dx$$$$:={\int }_0^{\pi }(\frac{\pi }{2}-x).ln\left(sin\left(x\right)\right)dx$$$$:=\frac{\pi }{2}{\int }_0^{\pi }ln\left(sin\left(x\right)\right)dx-{\int }_0^{\pi }xln\left(sin\left(x\right)\right)dx$$$$:=\frac{\pi }{2}(-\pi ln\left(2\right))-\mathrm{J}......\left(1\right)$$$$\mathrm{J}:={\int }_0^{\pi }(\pi -x)ln\left(sin\left(x\right)\right)dx$$$$:=\pi (-\pi ln\left(2\right))-\mathrm{J}$$$$\therefore \mathrm{J}:=\frac{-{\pi }^2}{2}ln\left(2\right).......\left(2\right)$$$$\left(2\right)\Rrightarrow \left(1\right):\mathrm{I}=0.........◼$$$$$$

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