$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+\mathrm{b}−\mathrm{x}\right)\mathrm{dx} \\$$$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{3sin}\left(\mathrm{x}\right)−\mathrm{2sin}^{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{sin}\left(\mathrm{x}\right)+\mathrm{cos}\left(\mathrm{x}\right)} \\$$$$\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} {f}\left({x}\right){dx}=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} {f}\left(\frac{\pi}{\mathrm{2}}−{x}\right){dx}=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{3}{cos}\left({x}\right)−\mathrm{2}{cos}^{\mathrm{3}} \left({x}\right)}{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx} \\$$$$\mathrm{2}\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} {f}\left({x}\right){dx}=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{3}{sin}\left({x}\right)−\mathrm{2}{sin}^{\mathrm{2}} \left({x}\right)}{{cos}\left({x}\right)+{sin}\left({x}\right)}{dx}+\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{3}{cos}\left({x}\right)−\mathrm{2}{cos}^{\mathrm{3}} \left({x}\right)}{{cos}\left({x}\right)+{sin}\left({x}\right)}{dx} \\$$$$=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{3}{sin}\left({x}\right)+{cos}\left({x}\right)−\mathrm{2}{sin}^{\mathrm{3}} \left({x}\right)−\mathrm{2}{cos}^{\mathrm{3}} \left({x}\right)}{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx} \\$$$$=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{3}\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)−\mathrm{2}\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)\left({sin}^{\mathrm{2}} \left({x}\right)+{cos}^{\mathrm{2}} \left({x}\right)−{cos}\left({x}\right){sin}\left({x}\right)\right)}{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx} \\$$$$=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \mathrm{3}−\mathrm{2}\left(\mathrm{1}+{sin}\left({x}\right){cos}\left({x}\right)\right){dx} \\$$$$=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \mathrm{1}−\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right){dx}=\left[{x}+{cos}^{\mathrm{2}} \left({x}\right)\right]_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} =\frac{\pi}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{3}}{\mathrm{4}}=\frac{\pi−\mathrm{3}}{\mathrm{6}} \\$$$$\\$$