Question Number 87793 by M±th+et£s last updated on 06/Apr/20
$${show}\:{that} \\ $$$$\int{e}^{{sin}\left({x}\right)} \:{dx}= \\ $$$$−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\left[\:{cos}\left({x}\right)\ast\left({sin}\left({x}\right)\right)^{{n}+\mathrm{1}} \ast\left[\left({sin}\left({x}\right)\right)^{\mathrm{2}} \right]^{\left(\frac{−{n}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\right)} \ast\:\mathrm{2}{F}_{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}−{n}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\left({cos}\left({x}\right)\right)^{\mathrm{2}} \right]\:\right]+{c} \\ $$$$ \\ $$$${notice}\backslash\mathrm{2}{F}_{\mathrm{1}} \:{is}\:{special}\:{function}\:{called}\:{hypergeometric}\:{function} \\ $$
Answered by mind is power last updated on 06/Apr/20
$$\int{sin}^{{m}} \left({x}\right){dx} \\ $$$$=\int\frac{{u}^{{m}} }{\:\sqrt{\mathrm{1}−{u}^{\mathrm{2}} }}{du}\int\underset{{n}\geqslant\mathrm{0}} {\sum}{u}^{{m}} .\frac{\left(\mathrm{2}{n}\right)!{u}^{\mathrm{2}{n}} }{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{du}=\underset{{n}\geqslant\mathrm{0}} {\sum}\int\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{u}^{\mathrm{2}{n}+{m}} {du} \\ $$$$=\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }.\frac{{u}^{\mathrm{2}{n}+{m}+\mathrm{1}} }{\left(\mathrm{2}{n}+{m}+\mathrm{1}\right)}+{c} \\ $$$$={u}^{{m}+\mathrm{1}} \left(\frac{\mathrm{1}}{{m}+\mathrm{1}}+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{2}^{\mathrm{2}{n}} \underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+\frac{{m}+\mathrm{3}}{\mathrm{2}}\right)}\frac{{u}^{\mathrm{2}{n}} }{{n}!}\right) \\ $$$$={u}^{{m}+\mathrm{1}} \\ $$$$=\frac{{u}^{{m}+\mathrm{1}} }{{m}+\mathrm{1}}.\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right).\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+\frac{{m}+\mathrm{1}}{\mathrm{2}}\right)}{\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({k}+\frac{{m}+\mathrm{3}}{\mathrm{2}}\right)}.\frac{{u}^{\mathrm{2}{n}} }{{n}!} \\ $$$$=\frac{{u}^{{m}+\mathrm{1}} }{{m}+\mathrm{1}}.\mathrm{2}{F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{{m}+\mathrm{1}}{\mathrm{2}};\frac{{m}+\mathrm{3}}{\mathrm{2}};{u}^{\mathrm{2}} \right) \\ $$$${e}^{{sin}\left({u}\right)} =\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{{sin}^{{m}} \left({u}\right)}{{m}!} \\ $$$$\int{e}^{{sin}\left({u}\right)} {du}=\int\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{{sin}^{{m}} \left({u}\right)}{{m}!}{du} \\ $$$$=\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{m}!}\int{sin}^{{m}} \left({u}\right){du}=\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{{sin}^{{m}+\mathrm{1}} \left({u}\right)}{{m}!\left({m}+\mathrm{1}\right)}.\mathrm{2}{F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{{m}+\mathrm{1}}{\mathrm{2}};\frac{{m}+\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \left({u}\right)\right) \\ $$$$=\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{{sin}^{{m}+\mathrm{1}} \left({u}\right)}{\left({m}+\mathrm{1}\right)!}\mathrm{2}{F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{{m}+\mathrm{1}}{\mathrm{2}};\frac{{m}+\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \left({u}\right)\right)+{c} \\ $$$${may}\:{bee}\:{mistaks}\: \\ $$
Commented by M±th+et£s last updated on 06/Apr/20
$${may}\:{bee}\:{i}\:{make}\:{a}\:{mistake}\:{i}\:{will}\:{post}\:{my} \\ $$$${solution} \\ $$
Answered by M±th+et£s last updated on 06/Apr/20
$${e}^{{u}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:,\:{u}={sin}\left({x}\right)\Rightarrow\Rightarrow{e}^{{sin}\left({x}\right)} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({sin}\left({x}\right)\right)^{{n}} }{{n}!} \\ $$$${I}=\int{e}^{{sin}\left({x}\right)} {dx}=\int\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({sin}\left({x}\right)\right)^{{n}} }{{n}!}{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int\left({sin}\left({x}\right)\right)^{{n}} {dx} \\ $$$$\int\left({sin}\left({x}\right)\right)^{{n}} {dx}=−{cos}\left({x}\right)\ast\left({sin}\left({x}\right)\right)^{{n}+\mathrm{1}} \ast\left[{sin}\left({x}\right)^{\mathrm{2}} \right]^{\frac{−{n}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}} \ast\mathrm{2}{F}_{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}−{n}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\left({cos}\left({x}\right)\right)^{\mathrm{2}} \right]{o} \\ $$$$\int{e}^{{sin}\left({x}\right)} {dx}=−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\left[{cos}\left({x}\right)\ast{sin}\left({x}\right)^{{n}+\mathrm{1}} \ast\left[\left({sin}\left({x}\right)\right)^{\mathrm{2}} \right]^{\frac{−{n}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}} \ast\mathrm{2}{F}_{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}−{n}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\left({icos}\left({x}\right)^{\mathrm{2}} \right)\right]\right] \\ $$
Commented by M±th+et£s last updated on 06/Apr/20
$${there}\:{is}\:{a}\:{typo} \\ $$$${its}\:\left[\left({cos}\left({x}\right)\right)^{\mathrm{2}} \right]\:{not}\:\left[\left({icos}\left({x}\right)^{\mathrm{2}} \right)\right] \\ $$