Question Number 38127 by maxmathsup by imad last updated on 22/Jun/18 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{2}{x}} }{dx} \\ $$ Commented by abdo mathsup 649 cc last…
Question Number 38126 by maxmathsup by imad last updated on 22/Jun/18 $${calculate}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{4}{x}} }{dx}\:. \\ $$ Commented by math khazana by abdo last…
Question Number 38123 by maxmathsup by imad last updated on 22/Jun/18 $${f}\:{is}\:{a}\:{function}\:{positive}\:\:{and}\:{C}^{\mathrm{1}} \:\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\:\frac{{f}^{'} }{\mathrm{2}\sqrt{{f}}\sqrt{\mathrm{1}+{f}}}{dx} \\ $$$$\left.\mathrm{2}\right){let}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{x}^{\frac{{n}}{\mathrm{2}}} }{{x}\sqrt{\mathrm{1}+{x}^{{n}} }} \\ $$$${calculate}\:{A}_{{n}}…
Question Number 38124 by maxmathsup by imad last updated on 22/Jun/18 $${prove}\:{that}\:\:\int\:\:\:\:\:\:\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:={ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)\:+{c} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\:\sqrt{{a}+{x}^{\mathrm{2}} }}\:{with}\:{a}>\mathrm{0} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 38125 by maxmathsup by imad last updated on 22/Jun/18 $${let}\:\alpha>\mathrm{0}\:{find} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{e}^{−\alpha{x}} }{\:\sqrt{\mathrm{1}+{e}^{−\mathrm{2}\alpha{x}} }}{dx}\:. \\ $$ Commented by math khazana by…
Question Number 38122 by maxmathsup by imad last updated on 22/Jun/18 $${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{sinx}}{\:\sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}}{dx} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18 $${t}={cosx}\:\:\:{dt}=−{sinxdx}…
Question Number 38121 by maxmathsup by imad last updated on 22/Jun/18 $${let}\:{x}>\mathrm{0}\:{find}\:{F}\left({x}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\: \\ $$ Commented by prof Abdo imad last updated…
Question Number 38119 by maxmathsup by imad last updated on 22/Jun/18 $${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{4}} \sqrt{{x}−\mathrm{1}}} \\ $$ Commented by math khazana by abdo last updated…
Question Number 38120 by maxmathsup by imad last updated on 22/Jun/18 $${let}\:\:{n}\:{from}\:{N}\:{and} \\ $$$${find}\:{the}\:{value}\:{of}\:\:{A}_{{n}} =\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{{n}} \sqrt{{t}−\mathrm{1}}} \\ $$ Commented by math khazana by…
Question Number 38118 by maxmathsup by imad last updated on 21/Jun/18 $${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{{t}^{{a}} }{\mathrm{2}}}{dt}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} \left({na}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} \left(\mathrm{3}{n}+\mathrm{1}\right)}…