$\underset{-\pi /2}{\overset{\pi /2}{\int }}\left\{\log \left(\frac{1+Sinx}{1-Sinx}\right)+x^{3}Co{s}^{2007}x+x^{2}Si{n}^{9}x+Si{n}^{6}x\right\}dx$

$$\begin{gathered} \underset{-\mathrm{\pi }/2}{\overset{\mathrm{\pi }/2}{\int }}\left\{\log \left(\frac{1+\mathrm{Sinx}}{1-\mathrm{Sinx}}\right)+{\mathrm{x}}^3{\mathrm{Cos}}^{2007}\mathrm{x}+{\mathrm{x}}^2{\mathrm{Sin}}^9\mathrm{x}+{\mathrm{Sin}}^6\mathrm{x}\right\}\mathrm{dx} \\ \log \left(\frac{1+\mathrm{Sin} \mathrm{x}}{1- \mathrm{Sin} \mathrm{x}}\right), {\mathrm{x}}^3 {\cos }^{2007}\mathrm{x} {\mathrm{x}}^2 {\mathrm{Sin}}^9\mathrm{x} \mathrm{are} \mathrm{odd} \mathrm{functions}. \\ \mathrm{I}=0+0+0+\underset{-\mathrm{\pi }/2}{\overset{\mathrm{\pi }/2}{\int }} {\mathrm{Sin}}^6\mathrm{x} \mathrm{dx} \\ =2 \underset{0}{\overset{\mathrm{\pi }/2}{\int }}{\mathrm{Sin}}^6\mathrm{x} \mathrm{dx} ({\mathrm{Sin}}^6\mathrm{x}-\mathrm{even}) \\ =2\left[\frac{5}{6}.\frac{3}{4}.\frac{1}{2}.\frac{\mathrm{\pi }}{2}\right]=\frac{5\mathrm{\pi }}{16} \end{gathered}$$