Question Number 121466 by sdfg last updated on 08/Nov/20 | ||
$$\int\sqrt{{cos}\left({x}\right)\:{dx}} \\ $$ | ||
Commented by MJS_new last updated on 08/Nov/20 | ||
$$\int\sqrt{\mathrm{cos}\:{x}\:{dx}}\:\mathrm{or}\:\int\sqrt{\mathrm{cos}\:{x}}\:{dx}? \\ $$ | ||
Commented by sdfg last updated on 08/Nov/20 | ||
$$\int\sqrt{\mathrm{cos}\:\mathrm{x}\:}\mathrm{dx} \\ $$ | ||
Answered by MJS_new last updated on 08/Nov/20 | ||
$$\int\sqrt{\mathrm{cos}\:{x}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}}{\mathrm{2}}\:\rightarrow\:{dx}=\mathrm{2}{dt}\right] \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{cos}\:\mathrm{2}{t}}\:{dt}=\mathrm{2}\int\sqrt{\mathrm{2cos}^{\mathrm{2}} \:{t}\:−\mathrm{1}}\:{dt}= \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{t}}\:{dt}=\mathrm{2E}\:\left({t}\mid\mathrm{2}\right)\:= \\ $$$$=\mathrm{2E}\:\left(\frac{{x}}{\mathrm{2}}\mid\mathrm{2}\right)\:+{C} \\ $$ | ||
Commented by MJS_new last updated on 08/Nov/20 | ||
$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}=\int\sqrt{\mathrm{cos}\:\left({x}−\frac{\pi}{\mathrm{2}}\right)}\:{dx} \\ $$$$\Rightarrow \\ $$$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}=\mathrm{2E}\:\left(\frac{{x}}{\mathrm{2}}−\frac{\pi}{\mathrm{4}}\mid\mathrm{2}\right)\:+{C} \\ $$ | ||
Commented by MJS_new last updated on 08/Nov/20 | ||
$$\left(\mathrm{Elliptic}\:\mathrm{Integral}\right) \\ $$ | ||
Commented by peter frank last updated on 08/Nov/20 | ||
$$\int\sqrt{\mathrm{sin}\:\mathrm{x}}\:{dx}=? \\ $$ | ||
Commented by peter frank last updated on 08/Nov/20 | ||
$$\mathrm{thank}\:\mathrm{you} \\ $$ | ||