Question Number 97590 by bhuvanmahant last updated on 08/Jun/20
$$\mathrm{find}\:\mathrm{the}\:\mathrm{integration} \\ $$$$\int\sqrt{{sin}\left({x}\right)}{dx} \\ $$
Answered by smridha last updated on 09/Jun/20
$$\boldsymbol{{let}}\:..\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{u}}^{\mathrm{2}} \\ $$$$\boldsymbol{{so}}\:\int\mathrm{2}{u}^{\mathrm{2}} \left(\mathrm{1}−\boldsymbol{{u}}^{\mathrm{4}} \right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \boldsymbol{{du}} \\ $$$$=\mathrm{2}\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2}\boldsymbol{\mathrm{n}}\right)!}{\mathrm{4}^{\boldsymbol{{n}}} \left(\boldsymbol{{n}}!\right)^{\mathrm{2}} }\int\boldsymbol{{u}}^{\mathrm{4}\boldsymbol{{n}}+\mathrm{2}} \boldsymbol{{du}} \\ $$$$=\mathrm{2}\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2}\boldsymbol{{n}}\right)!}{\mathrm{4}^{\boldsymbol{{n}}} \left(\boldsymbol{{n}}!\right)^{\mathrm{2}} }.\frac{\left(\boldsymbol{{sinx}}\right)^{\frac{\mathrm{4}\boldsymbol{{n}}+\mathrm{3}}{\mathrm{2}}} }{\left(\mathrm{4}\boldsymbol{{n}}+\mathrm{3}\right)}\:+\boldsymbol{{c}} \\ $$