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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

∫(√(cos x)) dx= [t=(x/2) → dx=2dt] =2∫(√(cos 2t)) dt=2∫(√(2cos^2 t −1)) dt= =2∫(√(1−2sin^2 t)) dt=2E (t∣2) = =2E ((x/2)∣2) +C

$$\int\sqrt{\mathrm{cos}\:{x}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}}{\mathrm{2}}\:\rightarrow\:{dx}=\mathrm{2}{dt}\right] \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{cos}\:\mathrm{2}{t}}\:{dt}=\mathrm{2}\int\sqrt{\mathrm{2cos}^{\mathrm{2}} \:{t}\:−\mathrm{1}}\:{dt}= \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{t}}\:{dt}=\mathrm{2E}\:\left({t}\mid\mathrm{2}\right)\:= \\ $$$$=\mathrm{2E}\:\left(\frac{{x}}{\mathrm{2}}\mid\mathrm{2}\right)\:+{C} \\ $$

Commented by MJS_new last updated on 08/Nov/20

∫(√(sin x)) dx=∫(√(cos (x−(π/2)))) dx ⇒ ∫(√(sin x)) dx=2E ((x/2)−(π/4)∣2) +C

$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}=\int\sqrt{\mathrm{cos}\:\left({x}−\frac{\pi}{\mathrm{2}}\right)}\:{dx} \\ $$$$\Rightarrow \\ $$$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}=\mathrm{2E}\:\left(\frac{{x}}{\mathrm{2}}−\frac{\pi}{\mathrm{4}}\mid\mathrm{2}\right)\:+{C} \\ $$

Commented by MJS_new last updated on 08/Nov/20