Question Number 139524 by mnjuly1970 last updated on 28/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:……{advanced}\:\:{calculus}….. \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:{i}::\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\mathrm{2}{x}} −\frac{\mathrm{1}}{\mathrm{1}+{e}^{{x}} }\right)\frac{\mathrm{1}}{{x}}\:{dx}={log}\left(\frac{\mathrm{1}}{\:\sqrt{\pi}}\:\right) \\ $$$$\:{ii}::\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{1}+{e}^{−{x}} }\right)\frac{{e}^{−\mathrm{2}{x}} }{{x}}{dx}={log}\left(\frac{\sqrt{\pi}}{\mathrm{2}}\right) \\…
Question Number 139476 by aliibrahim1 last updated on 27/Apr/21 Answered by qaz last updated on 27/Apr/21 $${I}=\int\frac{{dx}}{{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)…\left({x}+{n}\right)} \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)…\left({x}+{n}\right)}=\frac{{A}_{\mathrm{0}} }{{x}}+\frac{{A}_{\mathrm{1}} }{{x}+\mathrm{1}}+\frac{{A}_{\mathrm{2}} }{{x}+\mathrm{2}}+…+\frac{{A}_{{n}} }{{x}+{n}} \\ $$$${A}_{\mathrm{0}}…
Question Number 139478 by mnjuly1970 last updated on 27/Apr/21 $$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:………\:{nice}\:…\:…\:…\:{calculus}…….. \\ $$$$\:\:\:\:\:\:\Phi:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{2}}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{24}} \\ $$$$\:\:\:\:\:\:\:{NOTE}\:::\:{li}_{\mathrm{2}} \left({z}\right)+{li}_{\mathrm{2}} \left(\mathrm{1}−{z}\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{6}}−{ln}\left({z}\right){ln}\left(\mathrm{1}−{z}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{Hence}\:::\:\:{li}_{\mathrm{2}}…
Question Number 139454 by aliibrahim1 last updated on 27/Apr/21 Answered by Dwaipayan Shikari last updated on 27/Apr/21 $$\int{x}^{\mathrm{2}} \frac{{tan}^{−\mathrm{1}} {x}}{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$$$=\int{tan}^{−\mathrm{1}} \left({x}\right)−\int\frac{{tan}^{−\mathrm{1}} \left({x}\right)}{{x}^{\mathrm{2}}…
Question Number 139445 by aliibrahim1 last updated on 27/Apr/21 Answered by mr W last updated on 27/Apr/21 $${u}={xy} \\ $$$$\frac{{du}}{{dx}}={x}\frac{{dy}}{{dx}}+{y} \\ $$$$\frac{{du}}{{dx}}+\mathrm{3}{u}^{\mathrm{2}} =\mathrm{0} \\ $$$$−\frac{{du}}{{u}^{\mathrm{2}}…
Question Number 8375 by tawakalitu last updated on 09/Oct/16 $$\mathrm{Evaluate}\::\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \sqrt{\mathrm{1}\:−\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$$$\mathrm{By}\:\mathrm{direct}\:\mathrm{integration}\:\mathrm{and}\:\mathrm{by}\:\mathrm{expanding} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{power}\:\mathrm{series}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 8366 by tawakalitu last updated on 09/Oct/16 $$\int\frac{\mathrm{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{4}}}\:\mathrm{dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 139432 by normabaru last updated on 27/Apr/21 Answered by Jme Eduardo last updated on 27/Apr/21 $$\left({a}\right)\:\:{if}\:\:{x}={t}^{\mathrm{2}} \:\:{and}\:\:{y}={t} \\ $$$$ \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{2}{t}^{\mathrm{4}} }{{t}^{\mathrm{4}}…
Question Number 139429 by bemath last updated on 27/Apr/21 $$\int\:\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{12x}\right)\:\sqrt[{\mathrm{5}\:}]{\mathrm{cos}\:\mathrm{6x}}\:\mathrm{dx}\:=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 139427 by Fikret last updated on 26/Apr/21 $$\underset{\:\sqrt{\mathrm{2}}} {\overset{\mathrm{3}\sqrt{\mathrm{2}}} {\int}}\sqrt{{x}^{\mathrm{2}} −\mathrm{2}}{dx}+\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}{dx}\:=? \\ $$ Answered by MJS_new last updated on 27/Apr/21…