Question Number 201224 by Calculusboy last updated on 02/Dec/23
$$\int\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{{sinx}}}}\boldsymbol{{dx}} \\ $$
Answered by witcher3 last updated on 02/Dec/23
$$\sqrt{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)}=\mathrm{hint}\::\mid\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mid \\ $$
Answered by Sutrisno last updated on 04/Dec/23
$$=\int\frac{\mathrm{1}}{\:\sqrt{{sin}^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{x}+{cos}^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{x}+\mathrm{2}{sin}\frac{\mathrm{1}}{\mathrm{2}}{xcos}\frac{\mathrm{1}}{\mathrm{2}}{x}}}{dx} \\ $$$$=\int\frac{\mathrm{1}}{\:\sqrt{\left({sin}\frac{\mathrm{1}}{\mathrm{2}}{x}+{cos}\frac{\mathrm{1}}{\mathrm{2}}{x}\right)^{\mathrm{2}} }}{dx} \\ $$$$=\int\frac{\mathrm{1}}{\:{sin}\frac{\mathrm{1}}{\mathrm{2}}{x}+{cos}\frac{\mathrm{1}}{\mathrm{2}}{x}}{dx} \\ $$$$=\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}{cos}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\pi}{\mathrm{4}}\right)}{dx} \\ $$$$=\int\frac{{sec}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\pi}{\mathrm{4}}\right)}{\:\sqrt{\mathrm{2}}}{dx} \\ $$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{2}}}{ln}\mid{sec}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\pi}{\mathrm{4}}\right)+{tan}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\pi}{\mathrm{4}}\right)\mid+{c} \\ $$
Commented by Calculusboy last updated on 04/Dec/23
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$