Question Number 216715 by Nadirhashim last updated on 16/Feb/25
$$\:\:\:\:\boldsymbol{{F}}{ind}\:\int\frac{\boldsymbol{{S}}{in}\left(\frac{\mathrm{5}{x}\:}{\mathrm{2}\:}\right)\:\:}{\boldsymbol{{S}}{in}\left(\frac{{x}\:}{\mathrm{2}\:}\right)\:\:\:\:\:}\:.\boldsymbol{{dx}}\:\:\: \\ $$
Answered by Frix last updated on 16/Feb/25
$$\int\frac{\mathrm{sin}\:\frac{\mathrm{5}{x}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}{dx}=\int\left(\mathrm{2cos}\:\mathrm{2}{x}\:+\mathrm{2cos}\:{x}\:+\mathrm{1}\right){dx}= \\ $$$$=\mathrm{sin}\:\mathrm{2}{x}\:+\mathrm{2sin}\:{x}\:+{x}+{C} \\ $$
Answered by mehdee7396 last updated on 17/Feb/25
$$\frac{{sin}\left(\frac{\mathrm{5}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}=\frac{{sin}\mathrm{2}{xcos}\left(\frac{{x}}{\mathrm{2}}\right)+{sin}\left(\frac{{x}}{\mathrm{2}}\right){cos}\mathrm{2}{x}}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\frac{\mathrm{4}{sin}\frac{{x}}{\mathrm{2}}{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}{cosx}}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}+{cos}\mathrm{2}{x} \\ $$$$=\mathrm{4}{cosxcos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+{cos}\mathrm{2}{x} \\ $$$$=\mathrm{2}{cosx}×\left(\mathrm{1}+{cosx}\right)+{cos}\mathrm{2}{x} \\ $$$$={cos}\mathrm{2}{x}+\mathrm{2}{cosx}+\mathrm{2}{cos}^{\mathrm{2}} {x} \\ $$$$=\mathrm{2}{cos}\mathrm{2}{x}+\mathrm{2}{cosx}+\mathrm{1} \\ $$$$\Rightarrow{I}={sin}\mathrm{2}{x}+\mathrm{2}{sinx}+{x}+{c}\: \\ $$$$ \\ $$