1-sin-x-dx-2-cos-x-2-dx-

Question Number 84607 by M±th+et£s last updated on 14/Mar/20

$$\left.\mathrm{1}\right)\int\sqrt{{sin}\left({x}\right)}\:{dx} \\ $$$$\left.\mathrm{2}\right)\int{cos}\left({x}^{\mathrm{2}} \right){dx} \\ $$$$ \\ $$

Commented by john santu last updated on 14/Mar/20

$$\mathrm{oo}\:\mathrm{yes}…\:\mathrm{it}\:\mathrm{typo}.\: \\ $$

Answered by MJS last updated on 14/Mar/20

∫(√(sin x)) dx= [transforming with trigonometric formulas] =∫(√(1−2sin^2 ((2x−π)/4))) dx= [t=((2x−π)/4) → dx=2dt] =2∫(√(1−2sin^2 t)) dt= =2E (t ∣2) = =2E (((2x−π)/4) ∣ 2) +C

$$\int\sqrt{\mathrm{sin}\:{x}}\:{dx}= \\ $$$$\:\:\:\:\:\left[\mathrm{transforming}\:\mathrm{with}\:\mathrm{trigonometric}\:\mathrm{formulas}\right] \\ $$$$=\int\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}}\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}\:\rightarrow\:{dx}=\mathrm{2}{dt}\right] \\ $$$$=\mathrm{2}\int\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{t}}\:{dt}= \\ $$$$=\mathrm{2E}\:\left({t}\:\:\mid\mathrm{2}\right)\:= \\ $$$$=\mathrm{2E}\:\left(\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}\:\mid\:\mathrm{2}\right)\:+{C} \\ $$

Commented by M±th+et£s last updated on 14/Mar/20

$${thank}\:{you}\:{sir} \\ $$

Answered by MJS last updated on 14/Mar/20

∫cos x^2 dx= [t=(√(2/π))x → dx=(√(π/2))dt] =(√(π/2))∫cos ((π/2)t^2 ) dt= this is the Fresnell Integral =(√((π/2) ))C (t) =(√(π/2)) C ((√(2/π))x) +C

$$\int\mathrm{cos}\:{x}^{\mathrm{2}} \:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\frac{\mathrm{2}}{\pi}}{x}\:\rightarrow\:{dx}=\sqrt{\frac{\pi}{\mathrm{2}}}{dt}\right] \\ $$$$=\sqrt{\frac{\pi}{\mathrm{2}}}\int\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}{t}^{\mathrm{2}} \right)\:{dt}= \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{Fresnell}\:\mathrm{Integral} \\ $$$$=\sqrt{\frac{\pi}{\mathrm{2}}\:}\mathrm{C}\:\left({t}\right)\:=\sqrt{\frac{\pi}{\mathrm{2}}}\:\mathrm{C}\:\left(\sqrt{\frac{\mathrm{2}}{\pi}}{x}\right)\:+{C} \\ $$

Commented by M±th+et£s last updated on 14/Mar/20