Question Number 141367 by cesarL last updated on 18/May/21
$$\int\left(\sqrt{{cosx}\centerdot{senx}}\right){dx} \\ $$
Answered by MJS_new last updated on 18/May/21
![if sen x =sin x ∫(√(cos x sin x)) dx=((√2)/2)∫(√(sin 2x)) dx= [t=x−(π/4) → dx=dt] =((√2)/2)∫(√(cos 2t)) dt=((√2)/2)∫(√(1−2sin^2 t)) dt= =((√2)/2)E (t∣2) = =((√2)/2)E (x−(π/4)∣2) +C](https://www.tinkutara.com/question/Q141385.png)
$$\mathrm{if}\:\mathrm{sen}\:{x}\:=\mathrm{sin}\:{x} \\ $$$$\int\sqrt{\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}}\:{dx}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\sqrt{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}={x}−\frac{\pi}{\mathrm{4}}\:\rightarrow\:{dx}={dt}\right] \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\sqrt{\mathrm{cos}\:\mathrm{2}{t}}\:{dt}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{t}}\:{dt}= \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{E}\:\left({t}\mid\mathrm{2}\right)\:= \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{E}\:\left({x}−\frac{\pi}{\mathrm{4}}\mid\mathrm{2}\right)\:+{C} \\ $$