Question Number 35684 by prof Abdo imad last updated on 22/May/18 $${calculate}\:{I}\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{\mathrm{2}{t}} \:{ln}\left(\mathrm{1}+{e}^{{t}} \right){dt} \\ $$ Commented by prof Abdo imad last updated…
Question Number 35682 by prof Abdo imad last updated on 22/May/18 $${let}\:{F}\left({x}\right)\:=\:\int_{{x}\:+\mathrm{1}} ^{{x}^{\mathrm{2}} \:+\mathrm{1}} \:\:\:{arctan}\left(\mathrm{1}+{t}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\frac{\partial{F}}{\partial{x}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right)\:. \\ $$ Commented by tanmay.chaudhury50@gmail.com…
Question Number 35681 by prof Abdo imad last updated on 22/May/18 $${find}\:\:\int\:{arctan}\left({x}\right){dx} \\ $$ Commented by prof Abdo imad last updated on 23/May/18 $${let}\:{integrate}\:{by}\:{parts}\:{u}^{'} \:=\mathrm{1}\:{and}\:{v}={arctan}\left({x}\right)…
Question Number 35680 by prof Abdo imad last updated on 22/May/18 $${by}\:{using}\:{residus}\:{theorem}\:{calculate} \\ $$$${W}_{{n}} \:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{cos}^{\mathrm{2}{n}} {t}\:{dt}\:\:\left(\:\:{wallis}\:{integal}\right)\:{n}\:{integr} \\ $$$${natural}\:. \\ $$ Terms of Service…
Question Number 35678 by abdo imad last updated on 21/May/18 $${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{existencte}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right). \\…
Question Number 35676 by abdo imad last updated on 21/May/18 $${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{ch}^{\mathrm{4}} {t}\:{dt} \\ $$ Commented by abdo mathsup 649 cc last updated on…
Question Number 35677 by abdo imad last updated on 21/May/18 $${find}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{e}^{−\mathrm{2}{t}} {cos}\left({t}+\frac{\pi}{\mathrm{4}}\right){dx}. \\ $$ Commented by prof Abdo imad last updated on 23/May/18…
Question Number 35675 by abdo imad last updated on 21/May/18 $${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{x}}{{e}^{{x}} \:−\mathrm{1}}{dx}\:.. \\ $$ Commented by prof Abdo imad last updated on 25/May/18…
Question Number 166725 by cortano1 last updated on 26/Feb/22 $$\:\:\:\int\:\frac{\mathrm{dx}}{\mathrm{3}+\mathrm{tan}\:\mathrm{x}}=? \\ $$ Commented by MJS_new last updated on 26/Feb/22 $$\int\frac{{dx}}{\mathrm{3}+\mathrm{tan}\:{x}}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{tan}\:{x}\:\rightarrow\:{dx}=\frac{{dt}}{{t}^{\mathrm{2}} +\mathrm{1}}\right]= \\ $$$$=\int\frac{{dt}}{\left({t}+\mathrm{3}\right)\left({t}^{\mathrm{2}}…
Question Number 101192 by bemath last updated on 01/Jul/20 $$\int\:\frac{{x}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}\: \\ $$ Commented by bobhans last updated on 01/Jul/20 $$\mathrm{set}\:{z}\:=\:\frac{\pi}{\mathrm{2}}−{x}\:\Rightarrow\:{dx}\:=\:−{dz} \\ $$$${I}=\int\:\frac{\frac{\pi}{\mathrm{2}}−{z}}{\mathrm{1}+\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−{z}\right)}\:\left(−{dz}\right)\:=\:\int\:\frac{{z}−\frac{\pi}{\mathrm{2}}}{\mathrm{1}+\mathrm{cos}\:{z}}\:{dz} \\ $$$$=\:\int\:\left(\mathrm{z}−\frac{\pi}{\mathrm{2}}\right)\:\mathrm{d}\left(\mathrm{tan}\:\left(\frac{\mathrm{z}}{\mathrm{2}}\right)\right)\:=\:\left(\mathrm{z}−\frac{\pi}{\mathrm{2}}\right)\mathrm{tan}\:\left(\frac{\mathrm{z}}{\mathrm{2}}\right)+\mathrm{2}\:\mathrm{ln}\left(\mathrm{cos}\:\left(\frac{\mathrm{z}}{\mathrm{2}}\right)\right)\:+\:\mathrm{c} \\…