Question Number 102064 by bramlex last updated on 06/Jul/20
$$\int\:\frac{\mathrm{sin}\:\mathrm{3x}}{\mathrm{cos}\:\mathrm{5x}.\:\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\:? \\ $$
Answered by Dwaipayan Shikari last updated on 06/Jul/20
$$\int\frac{{sin}\mathrm{3}{x}−{sin}\mathrm{7}{x}}{{cos}\mathrm{2}{xcos}\mathrm{5}{x}}+\frac{{sin}\mathrm{7}{x}}{{cos}\mathrm{2}{xcos}\mathrm{5}{x}}=\int−\mathrm{2}{tan}\mathrm{2}{xdx}+\int\frac{{sin}\mathrm{5}{xcos}\mathrm{2}{x}+{cos}\mathrm{5}{xsin}\mathrm{2}{x}}{{cos}\mathrm{2}{xcos}\mathrm{5}{x}}{dx} \\ $$$$\int−{tan}\mathrm{2}{xdx}+\int{tan}\mathrm{5}{xdx}=\frac{\mathrm{1}}{\mathrm{2}}{log}\left({cos}\mathrm{2}{x}\right)−\frac{\mathrm{1}}{\mathrm{5}}{log}\left({cos}\mathrm{5}{x}\right)+{Constant} \\ $$