Question Number 6820 by Tawakalitu. last updated on 29/Jul/16 $${The}\:{position}\:{vector}\:{of}\:{the}\:{point}\:{P}\:\:{at}\:{time}\:{t}\:{is}\:{given}\:{by}\: \\ $$$$\left(\alpha{tant}\right){i}\:+\:\left(\alpha{sect}\right){k}\:{where}\:\alpha\:{is}\:{positive}\:{constant}\:{and} \\ $$$$\mathrm{0}\leqslant{t}\leqslant\frac{\Pi}{\mathrm{2}}.\:\:{Show}\:{that}\:{the}\:{velocity}\:{and}\:{acceleration}\:{of}\:{P}\:\:{when} \\ $$$${t}\:=\:\mathrm{0}\:{are}\:{at}\:{right}\:{angle}\:{to}\:{each}\:{other}.\:{If}\:{A}\:{is}\:{the}\:{point}\:{with}\: \\ $$$${position}\:{vector}\:\alpha{j},\:{obtain}\:{the}\:{vector}\:{equation}\:{for}\:{the}\:{straight}\: \\ $$$${line}\:{AP}\:{at}\:{time}\:{t}.\:\:{If}\:{the}\:{point}\:{Q}\:{divides}\:{AP}\:\:{internally}\:{in}\:{the} \\ $$$${ratio}\:\left({cost}\right)\::\:\left(\mathrm{1}\:−\:{cost}\right).\:{Show}\:{that}\:{the}\:{acceleration}\:{of}\:{the}\:{point} \\ $$$${Q}\:{is}\:{constant}\:{in}\:{magnitude}\:{and}\:{is}\:{always}\:{directed}\:{towards}\:{a}\: \\…
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Question Number 137891 by bobhans last updated on 07/Apr/21 $$\underset{\mathrm{1}} {\overset{\mathrm{5}} {\int}}\:\frac{\mathrm{1}+\sqrt[{\mathrm{5}}]{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{5}\right)}\:\mathrm{cos}\:\pi{x}}{{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{10}}\:{dx}\: \\ $$ Answered by bemath last updated on 07/Apr/21 Terms of Service…
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Question Number 72346 by aliesam last updated on 27/Oct/19 Commented by mathmax by abdo last updated on 27/Oct/19 $${I}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}^{\mathrm{10}} +{x}^{\mathrm{6}} \:+{x}^{\mathrm{4}} \:+\mathrm{1}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\:\:\:{by}\:{psrts}\:{u}^{'}…
Question Number 6811 by Tawakalitu. last updated on 28/Jul/16 $${Let}\:\:\:\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\:{log}_{\mathrm{2}} {k}\:\:=\:\:\mathrm{1994} \\ $$$${find}\:{the}\:{positive}\:{integer}\:{n}\:? \\ $$ Commented by Yozzii last updated on 29/Jul/16 $$\underset{{k}=\mathrm{1}}…
Question Number 6810 by Tawakalitu. last updated on 28/Jul/16 $${Up}\:{to}\:{which}\:{prime}\:{number}\:{should}\:\:\:\mathrm{9}\:\:\:{divide}\:\:\:\mathrm{9631}\:\:\:{to}\:{ensure}\:{that} \\ $$$${this}\:{number}\:{is}\:{a}\:{prime}\:{number}\:{or}\:{a}\:{composite}\:{number}\:? \\ $$ Commented by Rasheed Soomro last updated on 29/Jul/16 $$\mathrm{97} \\ $$$${The}\:{largest}\:{prime}\:{number}\:{not}\:{greater}\:{than}\:\sqrt{\mathrm{9631}}…
Question Number 72344 by Rio Michael last updated on 27/Oct/19 $${prove}\:{that}\:\:{e}^{{lnx}} \:=\:{x} \\ $$$${or}\:\:{a}^{{log}_{{a}} {x}} \:=\:{x} \\ $$ Commented by Prithwish sen last updated on…
Question Number 6809 by Tawakalitu. last updated on 28/Jul/16 $${If}\:\:{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{y}^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{z}^{\frac{\mathrm{1}}{\mathrm{3}}} \:\:=\:\:\mathrm{0},\:\:{Then}\:\:{pove}\:{that}\: \\ $$$$\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{3}} \:\:=\:\mathrm{3}{xyz} \\ $$ Commented by Yozzii last updated on 29/Jul/16…
Question Number 137876 by Bekzod Jumayev last updated on 07/Apr/21 Answered by MJS_new last updated on 07/Apr/21 $$\int\frac{{dx}}{\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} }= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}}{\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} }\:\rightarrow\:{dx}=−\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{4}/\mathrm{3}}…