Question Number 7385 by Tawakalitu. last updated on 25/Aug/16
$$\int{sin}\left(\mathrm{101}{x}\right)\:.\:{sin}^{\mathrm{90}} \left({x}\right)\:{dx} \\ $$
Commented by Yozzia last updated on 26/Aug/16
$${Let}\:{I}\left({n}\right)=\int{e}^{\mathrm{101}{ix}} {sin}^{{n}} {xdx}\:\:\:\left({n}\in\mathbb{Z}^{\geqslant} \:{i}=\sqrt{−\mathrm{1}}\right) \\ $$$${I}\left({n}\right)=\int{e}^{\mathrm{101}{ix}} {sin}^{\mathrm{2}} {xsin}^{{n}−\mathrm{2}} {xdx} \\ $$$${I}\left({n}\right)=\int{e}^{\mathrm{101}{xi}} \left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right){sin}^{{n}−\mathrm{2}} {xdx} \\ $$$${I}\left({n}\right)={I}\left({n}−\mathrm{2}\right)−\int{e}^{\mathrm{101}{ix}} {cosx}\left\{{cosxsin}^{{n}−\mathrm{2}} {x}\right\}{dx} \\ $$$${I}\left({n}\right)={I}\left({n}−\mathrm{2}\right)−\left[{e}^{\mathrm{101}{ix}} {cosx}\frac{\mathrm{1}}{{n}−\mathrm{1}}{sin}^{{n}−\mathrm{1}} {x}−\frac{\mathrm{1}}{{n}−\mathrm{1}}\int{sin}^{{n}−\mathrm{1}} {xe}^{\mathrm{101}{ix}} \left\{\mathrm{101}{icosx}−{sinx}\right\}{dx}\right. \\ $$$${I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{101}{i}}{{n}−\mathrm{1}}\int{e}^{\mathrm{101}{ix}} {sin}^{{n}−\mathrm{1}} {xcosxdx}−\frac{\mathrm{1}}{{n}−\mathrm{1}}\int{e}^{\mathrm{101}{ix}} {sin}^{{n}} {xdx}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$${I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{101}{i}}{{n}−\mathrm{1}}\int{e}^{\mathrm{101}{ix}} {sin}^{{n}−\mathrm{1}} {xcosxdx}−\frac{\mathrm{1}}{{n}−\mathrm{1}}{I}\left({n}\right)−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\int{e}^{\mathrm{101}{ix}} {sin}^{{n}−\mathrm{1}} {xcosxdx}={e}^{\mathrm{101}{ix}} \frac{\mathrm{1}}{{n}}{sin}^{{n}} {x}−\frac{\mathrm{101}{i}}{{n}}\int{e}^{\mathrm{101}{ix}} {sin}^{{n}} {xdx} \\ $$$$\int{e}^{\mathrm{101}{ix}} {sin}^{{n}−\mathrm{1}} {xcosxdx}={e}^{\mathrm{101}{ix}} \frac{\mathrm{1}}{{n}}{sin}^{{n}} {x}−\frac{\mathrm{101}{i}}{{n}}{I}\left({n}\right) \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$${I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{101}{i}}{{n}−\mathrm{1}}\left\{\frac{\mathrm{1}}{{n}}{e}^{\mathrm{101}{ix}} {sin}^{{n}} {x}−\frac{\mathrm{101}{i}}{{n}}{I}\left({n}\right)\right\}−\frac{\mathrm{1}}{{n}−\mathrm{1}}{I}\left({n}\right)−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{n}−\mathrm{1}}\right){I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{10201}}{{n}\left({n}−\mathrm{1}\right)}{I}\left({n}\right)+\frac{\mathrm{101}{ie}^{\mathrm{101}{ix}} {sin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{n}−\mathrm{1}}−\frac{\mathrm{10201}}{{n}\left({n}−\mathrm{1}\right)}\right){I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{101}{ie}^{\mathrm{101}{ix}} {sin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$\frac{{n}^{\mathrm{2}} −\mathrm{10201}}{{n}\left({n}−\mathrm{1}\right)}{I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)=\frac{\mathrm{101}{ie}^{\mathrm{101}{ix}} {sin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$\Rightarrow{Im}\left\{\frac{{n}^{\mathrm{2}} −\mathrm{10201}}{{n}\left({n}−\mathrm{1}\right)}{I}\left({n}\right)−{I}\left({n}−\mathrm{2}\right)\right\}={Im}\left\{\frac{\mathrm{101}{ie}^{\mathrm{101}{ix}} {sin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}}\right\} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$${I}\left({n}\right)=\int{e}^{\mathrm{101}{ix}} {sin}^{{n}} {xdx}=\int{cos}\mathrm{101}{xsin}^{{n}} {xdx}+{i}\int{sin}\mathrm{101}{xsin}^{{n}} {xdx} \\ $$$$\Rightarrow{Im}\left\{{I}\left({n}\right)\right\}={Q}\left({n}\right)=\int{sin}\mathrm{101}{xsin}^{{n}} {xdx} \\ $$$$\therefore{Im}\left\{{I}\left({n}−\mathrm{2}\right)\right\}={Q}\left({n}−\mathrm{2}\right)=\int{sin}\mathrm{101}{xsin}^{{n}−\mathrm{2}} {xdx} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$${Im}\left\{\frac{\mathrm{101}{ie}^{\mathrm{101}{ix}} {sin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{e}^{\mathrm{101}{ix}} {cosxsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}}\right\} \\ $$$$=\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{cosxsin}\mathrm{101}{xsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\therefore\frac{{n}^{\mathrm{2}} −\mathrm{10201}}{{n}\left({n}−\mathrm{1}\right)}{Q}\left({n}\right)−{Q}\left({n}−\mathrm{2}\right)={f}\left({n},{x}\right) \\ $$$${where}\:{f}\left({n},{x}\right)=\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{{n}} {x}}{{n}\left({n}−\mathrm{1}\right)}−\frac{{cosxsin}\mathrm{101}{xsin}^{{n}−\mathrm{1}} {x}}{{n}−\mathrm{1}}. \\ $$$$\therefore\left({n}^{\mathrm{2}} −\mathrm{10201}\right){Q}\left({n}\right)−{n}\left({n}−\mathrm{1}\right){Q}\left({n}−\mathrm{2}\right)={n}\left({n}−\mathrm{1}\right){f}\left({n},{x}\right)\:\:\left({n}\geqslant\mathrm{2}\right) \\ $$$$\left({n}−\mathrm{101}\right)\left({n}+\mathrm{101}\right){Q}\left({n}\right)−{n}\left({n}−\mathrm{1}\right){Q}\left({n}−\mathrm{2}\right)={n}\left({n}−\mathrm{1}\right){f}\left({n},{x}\right) \\ $$$${Q}\left({n}\right)={g}\left({n}\right)\left\{{Q}\left({n}−\mathrm{2}\right)+{f}\left({n},{x}\right)\right\}\:\: \\ $$$${where}\:{g}\left({n}\right)=\frac{{n}\left({n}−\mathrm{1}\right)}{{n}^{\mathrm{2}} −\mathrm{10201}}\:\:\:\left({n}\neq\mathrm{101}\right). \\ $$$${For}\:{say}\:{n}=\mathrm{8}, \\ $$$${Q}\left(\mathrm{2}\right)={g}\left(\mathrm{2}\right)\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)\right\} \\ $$$${Q}\left(\mathrm{4}\right)={g}\left(\mathrm{4}\right)\left\{{Q}\left(\mathrm{2}\right)+{f}\left(\mathrm{4},{x}\right)\right\} \\ $$$$={g}\left(\mathrm{4}\right){g}\left(\mathrm{2}\right)\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)\right\}+{g}\left(\mathrm{4}\right){f}\left(\mathrm{4},{x}\right) \\ $$$${Q}\left(\mathrm{4}\right)={g}\left(\mathrm{4}\right){g}\left(\mathrm{2}\right){Q}\left(\mathrm{0}\right)+{g}\left(\mathrm{4}\right){g}\left(\mathrm{2}\right){f}\left(\mathrm{2},{x}\right)+{g}\left(\mathrm{4}\right){f}\left(\mathrm{4},{x}\right) \\ $$$${Q}\left(\mathrm{6}\right)={g}\left(\mathrm{6}\right)\left\{{Q}\left(\mathrm{4}\right)+{f}\left(\mathrm{6},{x}\right)\right\} \\ $$$$={g}\left(\mathrm{6}\right){g}\left(\mathrm{4}\right){g}\left(\mathrm{2}\right)\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)\right\}+{g}\left(\mathrm{6}\right){g}\left(\mathrm{4}\right){f}\left(\mathrm{4},{x}\right)+{g}\left(\mathrm{6}\right){f}\left(\mathrm{6},{x}\right) \\ $$$${Q}\left(\mathrm{8}\right)={g}\left(\mathrm{8}\right)\left\{{Q}\left(\mathrm{6}\right)+{f}\left(\mathrm{8},{x}\right)\right\} \\ $$$$=\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{4}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)\right\}+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right)}\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{4}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}{f}\left(\mathrm{4},{x}\right)+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right){g}\left(\mathrm{4}\right)}\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{4}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}{f}\left(\mathrm{6},{x}\right)+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right){g}\left(\mathrm{4}\right){g}\left(\mathrm{6}\right)}\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{4}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}{f}\left(\mathrm{8},{x}\right)\:\: \\ $$
Commented by Tawakalitu. last updated on 25/Aug/16
$${Wow}\:,\:{thank}\:{you}\:{sir}.\:{God}\:{bless}\:{you}. \\ $$$$ \\ $$
Commented by Tawakalitu. last updated on 26/Aug/16
$${I}\:{really}\:{appreciate}\:{sir}. \\ $$
Commented by Tawakalitu. last updated on 26/Aug/16
$${I}\:{really}\:{appreciate}\:{sir}. \\ $$
Commented by Yozzia last updated on 26/Aug/16
$${Q}\left(\mathrm{8}\right)=\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{4}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right)}{f}\left(\mathrm{4},{x}\right)+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right){g}\left(\mathrm{4}\right)}{f}\left(\mathrm{6},{x}\right)+\frac{\mathrm{1}}{{g}\left(\mathrm{2}\right){g}\left(\mathrm{4}\right){g}\left(\mathrm{6}\right)}{f}\left(\mathrm{8},{x}\right)\right\} \\ $$$$ \\ $$$$\therefore\:{For}\:{n}=\mathrm{90}, \\ $$$${Q}\left(\mathrm{90}\right)=\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{45}} {\prod}}{g}\left(\mathrm{2}{i}\right)\right\}\left\{{Q}\left(\mathrm{0}\right)+{f}\left(\mathrm{2},{x}\right)+\underset{{k}=\mathrm{2}} {\overset{\mathrm{45}} {\sum}}\frac{{f}\left(\mathrm{2}{k},{x}\right)}{\underset{{i}=\mathrm{1}} {\overset{{k}−\mathrm{1}} {\prod}}{g}\left(\mathrm{2}{i}\right)}\right\} \\ $$$${Q}\left(\mathrm{0}\right)=\int{sin}\mathrm{101}{xsin}^{\mathrm{0}} {xdx}=\frac{−\mathrm{1}}{\mathrm{101}}{cos}\mathrm{101}{x}+{c} \\ $$$$\therefore\:{Q}\left(\mathrm{90}\right)=\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{45}} {\prod}}\frac{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}{\mathrm{4}{i}^{\mathrm{2}} −\mathrm{10201}}\right\}\left\{\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{\mathrm{2}} {x}}{\mathrm{2}}−\frac{{sin}\mathrm{2}{x}}{\mathrm{2}}{sin}\mathrm{101}{x}−\frac{{cosx}}{\mathrm{101}}+\underset{{k}=\mathrm{2}} {\overset{\mathrm{45}} {\sum}}\left[\frac{\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{\mathrm{2}{i}} {x}}{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}−\frac{{cosxsin}\mathrm{101}{xsin}^{\mathrm{2}{i}−\mathrm{1}} {x}}{\mathrm{2}{i}−\mathrm{1}}}{\underset{{i}=\mathrm{1}} {\overset{{k}−\mathrm{1}} {\prod}}\frac{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}{\mathrm{4}{i}^{\mathrm{2}} −\mathrm{10201}}}\right]\right\}+{D} \\ $$$$\int{sin}\left(\mathrm{101}{x}\right){sin}^{\mathrm{90}} {xdx}=\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{45}} {\prod}}\frac{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}{\mathrm{4}{i}^{\mathrm{2}} −\mathrm{10201}}\right\}\left\{\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{\mathrm{2}} {x}}{\mathrm{2}}−\frac{{sin}\mathrm{2}{x}}{\mathrm{2}}{sin}\mathrm{101}{x}−\frac{{cosx}}{\mathrm{101}}+\underset{{k}=\mathrm{2}} {\overset{\mathrm{45}} {\sum}}\left[\frac{\frac{\mathrm{101}{cos}\mathrm{101}{xsin}^{\mathrm{2}{i}} {x}}{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}−\frac{{cosxsin}\mathrm{101}{xsin}^{\mathrm{2}{i}−\mathrm{1}} {x}}{\mathrm{2}{i}−\mathrm{1}}}{\underset{{i}=\mathrm{1}} {\overset{{k}−\mathrm{1}} {\prod}}\frac{\mathrm{2}{i}\left(\mathrm{2}{i}−\mathrm{1}\right)}{\mathrm{4}{i}^{\mathrm{2}} −\mathrm{10201}}}\right]\right\}+{D} \\ $$$$ \\ $$
Commented by Tawakalitu. last updated on 26/Aug/16
$${I}\:{really}\:{appreciate}\:{sir}. \\ $$$${God}\:{bless}\:{you} \\ $$