Question Number 40152 by maxmathsup by imad last updated on 16/Jul/18 $${let}\:\:{f}\left({x}\right)\:=\:\:\int_{−\mathrm{1}} ^{{x}} \:\:\:\:\frac{{e}^{{t}} }{\:\sqrt{\mathrm{1}−{e}^{{t}} }}{dt}\:\:\:{with}\:{x}<\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\int_{−\mathrm{1}} ^{\mathrm{0}} \:\:\frac{{e}^{{t}} }{\:\sqrt{\mathrm{1}−{e}^{{t}} }}{dt} \\…
Question Number 40151 by maxmathsup by imad last updated on 16/Jul/18 $${let}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left({xsint}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{u}\:\in{R}\:\:\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cosu}\leqslant\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{24}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\leqslant{F}\left({x}\right)\leqslant\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right) \\…
Question Number 40150 by maxmathsup by imad last updated on 16/Jul/18 $${let}\:{f}_{{n}} \left({x}\right)\:=\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{{n}} \right)^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} }\:\:\:{ddfined}\:{on}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}_{{n}} \rightarrow^{{cs}} \:{f}\:\left({n}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}_{{n}} \left({x}\right){dx}…
Question Number 40148 by maxmathsup by imad last updated on 16/Jul/18 $${let}\:\:{f}\left({x}\right)=\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\:\:\frac{{k}^{\mathrm{3}}…
Question Number 40149 by maxmathsup by imad last updated on 16/Jul/18 $${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\:\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\:\sqrt{{n}+\mathrm{4}{k}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$ Commented by math khazana by…
Question Number 40147 by maxmathsup by imad last updated on 16/Jul/18 $${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\sqrt{{x}^{\mathrm{3}} \left(\mathrm{2}−{x}\right)}{dx} \\ $$ Commented by math khazana by abdo last updated…
Question Number 40145 by maxmathsup by imad last updated on 16/Jul/18 $${calculate}\:\int_{−\mathrm{7}} ^{−\mathrm{3}} \:\:\:\frac{\left({x}−\mathrm{1}\right){dx}}{\:\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{3}}} \\ $$ Commented by math khazana by abdo last updated…
Question Number 40146 by maxmathsup by imad last updated on 16/Jul/18 $${find}\:\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}}\:+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$ Commented by maxmathsup by imad last updated…
Question Number 171213 by MikeH last updated on 10/Jun/22 $$\mathrm{In}\:\mathrm{electricity},\:\mathrm{the}\:\mathrm{electrostatic}\:\mathrm{field} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{as}: \\ $$$${E}\:=\:\int_{\mathrm{0}} ^{\pi} \left[\frac{{a}^{\mathrm{2}} \sigma\:\mathrm{sin}\:\theta}{\mathrm{2}\epsilon\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} −\mathrm{2}{ax}\:\mathrm{cos}\theta}}\right]{d}\theta \\ $$$$\mathrm{where}\:{a},\sigma\:\mathrm{and}\:\epsilon\:\mathrm{are}\:\mathrm{constants}.\:\mathrm{Consider} \\ $$$$\mathrm{that}\:{x}>{a}\:\mathrm{and}\:\mathrm{show}\:\mathrm{that}\:{E}=\:\frac{{a}^{\mathrm{2}} \sigma}{\epsilon{x}} \\…
Question Number 40144 by maxmathsup by imad last updated on 16/Jul/18 $${find}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {x}\sqrt{{x}^{\mathrm{2}} \:−\mathrm{2}{x}\:+\mathrm{5}}\:{dx} \\ $$ Answered by maxmathsup by imad last updated on…