Question Number 141475 by bramlexs22 last updated on 19/May/21 $$\:{Find}\:{the}\:{smallest}\:{value}\:{of}\: \\ $$$$\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\:{among}\:{all}\:{values}\:{of} \\ $$$$\:{x}\:\&\:{y}\:{satisfying}\:\mathrm{3}{x}−{y}\:=\:\mathrm{20}\: \\ $$ Answered by MJS_new last updated on 19/May/21…
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Question Number 141378 by mnjuly1970 last updated on 18/May/21 $$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\frac{\left(\mathrm{5}{n}+\mathrm{2}\right)\left(\mathrm{5}{n}+\mathrm{3}\right)}{\left(\mathrm{5}{n}+\mathrm{1}\right)\left(\mathrm{5}{n}+\mathrm{4}\right)}\:=\varphi\: \\ $$$$\:\:\:\:\:\:\:\varphi:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$ Answered by Dwaipayan Shikari last updated on…
Question Number 141368 by bemath last updated on 18/May/21 $${A}\:{closed}\:{cylindrical}\:{can}\:{be}\:{is}\:{to}\:{hold} \\ $$$$\mathrm{1}\:{liters}\:{of}\:{liquid}\:.\:{How}\:{should}\:{we}\: \\ $$$${choose}\:{the}\:{height}\:{and}\:{radius}\: \\ $$$${to}\:{minimize}\:{the}\:{amount}\:{of} \\ $$$${material}\:{needed}\:{to}\:{manufacture} \\ $$$${the}\:{can}\:?\: \\ $$ Answered by TheSupreme…
Question Number 141346 by mnjuly1970 last updated on 17/May/21 Answered by qaz last updated on 17/May/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}^{\mathrm{2}} {x}}{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \frac{{y}^{\mathrm{2}}…
Question Number 75796 by ~blr237~ last updated on 17/Dec/19 $$\mathrm{let}\:\mathrm{be}\:\mathrm{a},\mathrm{b}\:\mathrm{such}\:\mathrm{as}\:\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} =\mathrm{ab} \\ $$$$\mathrm{find}\:\:\mathrm{out}\:\mathrm{Z}=\frac{\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} }{\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} }\:\:\mathrm{when}\:\:\mathrm{a}\neq\mathrm{0} \\ $$$$ \\ $$ Answered by mr…
Question Number 75795 by ~blr237~ last updated on 17/Dec/19 $$\mathrm{Find}\:\mathrm{out}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{argsh}\left(\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx} \\ $$ Commented by mathmax by abdo last updated on 18/Dec/19 $${let}\:{A}\:=\int_{\mathrm{0}} ^{\infty}…
Question Number 141328 by mnjuly1970 last updated on 17/May/21 $$……\:{Evaluate}: \\ $$$$\:\:\:\:\:\mathscr{F}\::=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{{n}+\mathrm{1}}\:=? \\ $$$$……. \\ $$ Answered by Dwaipayan Shikari last updated…
Question Number 141320 by mnjuly1970 last updated on 17/May/21 $$\:\:\:\:\:\:\:……{nice}\:……{calculuus}….. \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \frac{\mathscr{A}\:{rctan}\left({x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} }{dxdy}=\frac{\pi^{\mathrm{2}} \sqrt{\mathrm{2}}}{\mathrm{16}} \\ $$$$…..…
Question Number 75793 by ~blr237~ last updated on 17/Dec/19 $$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{arctanx}}{\mathrm{x}\sqrt{\mathrm{log2}}}\right)^{\mathrm{2}} \mathrm{dx}=\:\pi \\ $$ Commented by mathmax by abdo last updated on 18/Dec/19 $${let}\:{I}\:=\int_{\mathrm{0}}…