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sinx-dx-




Question Number 168556 by mokys last updated on 13/Apr/22
∫(√(sinx)) dx
$$\int\sqrt{{sinx}}\:{dx} \\ $$
Answered by MJS_new last updated on 13/Apr/22
this question keeps reappearing; I′ve given  the answer before...  ∫(√(sin x)) dx=∫(√(1−sin^2  ((2x−π)/4))) dx=       [t=((2x−π)/4) → dx=2dt]  =2∫(√(1−sin^2  t)) dt=2E (t∣2) =  =2E (((2x−π)/4)∣2) +C
$$\mathrm{this}\:\mathrm{question}\:\mathrm{keeps}\:\mathrm{reappearing};\:\mathrm{I}'\mathrm{ve}\:\mathrm{given} \\ $$$$\mathrm{the}\:\mathrm{answer}\:\mathrm{before}… \\ $$$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}=\int\sqrt{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}\:\rightarrow\:{dx}=\mathrm{2}{dt}\right] \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{t}}\:{dt}=\mathrm{2E}\:\left({t}\mid\mathrm{2}\right)\:= \\ $$$$=\mathrm{2E}\:\left(\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}\mid\mathrm{2}\right)\:+{C} \\ $$
Commented by peter frank last updated on 14/Apr/22
thanks
$$\mathrm{thanks} \\ $$

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