Question Number 150413 by saly last updated on 12/Aug/21 Commented by MJS_new last updated on 12/Aug/21 $$\mathrm{in}\:\mathrm{this}\:\mathrm{case}\:\mathrm{we}\:\mathrm{must}\:\mathrm{take}\:\mathrm{the}\:\mathrm{long}\:\mathrm{way}\:\mathrm{home}… \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\mathrm{arctan}\:{r}\:=−\frac{\mathrm{i}}{\mathrm{2}}\left(\mathrm{ln}\:\left(\mathrm{1}+\mathrm{i}{r}\right)\:−\mathrm{ln}\:\left(\mathrm{1}−\mathrm{i}{r}\right)\right) \\ $$$$\Rightarrow \\ $$$$\int\mathrm{arctan}\:\mathrm{sin}\:{x}\:{dx}=…
Question Number 84879 by sahnaz last updated on 17/Mar/20 $$\mathrm{e}^{\int\frac{\mathrm{2dx}}{\mathrm{xlnx}}} \\ $$ Commented by jagoll last updated on 17/Mar/20 $$\int\:\frac{\mathrm{2dx}}{\mathrm{x}\:\mathrm{lnx}}\:=\:\int\:\frac{\mathrm{2d}\left(\mathrm{lnx}\right)}{\mathrm{lnx}}\:=\:\int\:\mathrm{2}\frac{\mathrm{du}}{\mathrm{u}} \\ $$$$=\:\mathrm{2}\:\mathrm{ln}\:\mathrm{u}\:+\:\mathrm{c}\:,\:\left[\mathrm{u}\:=\:\mathrm{ln}\:\mathrm{x}\:\right] \\ $$$$=\:\mathrm{2ln}\left(\mathrm{lnx}\right)\:+\:\mathrm{2lnC}\:=\:\mathrm{2ln}\left(\mathrm{Clnx}\right) \\…
Question Number 19341 by tawa tawa last updated on 09/Aug/17 Commented by tawa tawa last updated on 09/Aug/17 $$\left(\mathrm{1}\right)\:\mathrm{Using}\:\mathrm{Stoke}'\mathrm{s}\:\mathrm{theorem},\:\mathrm{find}\:\:\oint\:\:\mathrm{y}^{\mathrm{2}} \mathrm{dx}\:+\:\mathrm{z}^{\mathrm{2}} \mathrm{dy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{dz}\:,\:\:\mathrm{where}\:\:\Gamma\:\mathrm{is}\:\mathrm{the}\:\mathrm{closed} \\ $$$$\mathrm{curve}\:\:\mathrm{A}\rightarrow\mathrm{B}\rightarrow\mathrm{C}\rightarrow\mathrm{A}\:\:\mathrm{and}\:\:\mathrm{A}\:=\:\left(\mathrm{1},\:\mathrm{0},\:\mathrm{0}\right),\:\:\mathrm{B}\:=\:\left(\mathrm{0},\:\mathrm{0},\:\mathrm{1}\right),\:\:\mathrm{C}\left(\mathrm{0},\:\mathrm{1},\:\mathrm{0}\right). \\…
Question Number 150410 by ajfour last updated on 12/Aug/21 Commented by ajfour last updated on 12/Aug/21 $${If}\:{length}\:{of}\:{cylinder}\:{and} \\ $$$${both}\:{width}\:{and}\:{length}\:{of} \\ $$$${wedge}\:{are}\:{equal},\:{and}\:{that} \\ $$$${their}\:{material}\:{densities} \\ $$$${are}\:{even}\:{equal},\:{and}\:{their}…
Question Number 84873 by john santu last updated on 17/Mar/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\left(\frac{\mathrm{x}!}{\mathrm{x}}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}} \\ $$ Commented by john santu last updated on 17/Mar/20 $$\mathrm{yes}.\:\mathrm{i}\:\mathrm{agree}\:\mathrm{sir}…
Question Number 150405 by liberty last updated on 12/Aug/21 Answered by EDWIN88 last updated on 12/Aug/21 Answered by ajfour last updated on 12/Aug/21 Commented by…
Question Number 150404 by mathdanisur last updated on 12/Aug/21 $$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{log}\left(\mathrm{x}\right)}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$ Answered by Ar Brandon last updated on 12/Aug/21 $$\Omega\left({a}\right)=\int_{\mathrm{0}}…
Question Number 84871 by tw000001 last updated on 17/Mar/20 $$\mathrm{If}\:\mathrm{you}\:\mathrm{know} \\ $$$$\left(\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}\right)^{\mathrm{2}} +\left(\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ca}}\right)^{\mathrm{2}} +\left(\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}\right)^{\mathrm{2}} =\mathrm{3}, \\…
Question Number 84868 by jagoll last updated on 17/Mar/20 $$\sqrt{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{3}\pi}{\mathrm{2}}−\mathrm{x}\right)}\:=\:−\mathrm{cos}\:\mathrm{x}+\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{sin}\:\left(\mathrm{x}−\pi\right) \\ $$ Answered by john santu last updated on 17/Mar/20 $$\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}−\mathrm{x}\right)\:=\:−\mathrm{sin}\:\mathrm{x} \\ $$$$\mathrm{sin}\:\left(\mathrm{x}−\pi\right)\:=\:−\mathrm{sin}\:\mathrm{x} \\…
Question Number 19332 by Tinkutara last updated on 09/Aug/17 $$\mathrm{Let}\:{S}_{{n}} \:=\:{n}^{\mathrm{2}} \:+\:\mathrm{20}{n}\:+\:\mathrm{12},\:{n}\:\mathrm{a}\:\mathrm{positive} \\ $$$$\mathrm{integer}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible} \\ $$$$\mathrm{values}\:\mathrm{of}\:{n}\:\mathrm{for}\:\mathrm{which}\:{S}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect} \\ $$$$\mathrm{square}? \\ $$ Commented by mrW1 last…