Question Number 65214 by naka3546 last updated on 26/Jul/19 $${Prove}\:\:{or}\:\:{disprove}\:\:\:{that}\:\:\:\mathrm{2}^{\mathrm{101}} \:\mid\:{n}^{{n}} \:−\:\mathrm{101}\:. \\ $$ Commented by naka3546 last updated on 26/Jul/19 $${n}\:\:{positive}\:\:{integer} \\ $$ Commented…
Question Number 130748 by Dwaipayan Shikari last updated on 28/Jan/21 $$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{11}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{13}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{19}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{21}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{29}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{31}^{\mathrm{3}} }+.. \\ $$ Answered by mindispower last…
Question Number 65212 by ajfour last updated on 26/Jul/19 Commented by ajfour last updated on 26/Jul/19 $${For}\:{P}\:{to}\:{be}\:{unique},\:{determine} \\ $$$${how}\:{a},{b},{c}\:{are}\:{related}. \\ $$ Answered by ajfour last…
Question Number 130744 by rs4089 last updated on 28/Jan/21 Answered by Dwaipayan Shikari last updated on 28/Jan/21 $${n}+\mathrm{1}=\sqrt{{n}^{\mathrm{2}} +\mathrm{2}{n}+\mathrm{1}}=\sqrt{\mathrm{1}+{n}\left({n}+\mathrm{2}\right)} \\ $$$$=\sqrt{\mathrm{1}+{n}\sqrt{\mathrm{1}+\left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right)}}=\sqrt{\mathrm{1}+{n}\sqrt{\mathrm{1}+\left({n}+\mathrm{1}\right)\sqrt{\mathrm{1}+\left({n}+\mathrm{2}\right)\sqrt{\mathrm{1}+…}}}} \\ $$$${n}=\mathrm{2} \\ $$$$\mathrm{3}=\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+\mathrm{3}\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}+…}}}}…
Question Number 130745 by greg_ed last updated on 28/Jan/21 $$\mathrm{hi},\:\mathrm{mr}\:\mathrm{tinku}\:\mathrm{tara}\:! \\ $$$$\mathrm{who}\:\mathrm{must}\:\mathrm{report}\:\mathrm{an}\:\mathrm{inappropriate}\:\mathrm{post} \\ $$$$\mathrm{on}\:\mathrm{this}\:\mathrm{forum}\:??? \\ $$$$\mathrm{should}\:\mathrm{any}\:\mathrm{member}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}\:\mathrm{have}\: \\ $$$$\mathrm{this}\:\mathrm{right}\:? \\ $$$$\mathrm{does}\:\mathrm{this}\:\mathrm{forum}\:\mathrm{have}\:\mathrm{a}\:\mathrm{moderator}\:? \\ $$ Commented by Ar…
Question Number 130742 by mohammad17 last updated on 28/Jan/21 Answered by Dwaipayan Shikari last updated on 28/Jan/21 $${z}=\mathrm{9}+\mathrm{3}{i}\:\:=\sqrt{\mathrm{9}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }\:{e}^{{itan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{3}}} =\mathrm{3}\sqrt{\mathrm{10}}{e}^{{itan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{3}}} \\ $$$${z}=\mathrm{7}+\mathrm{2}{i}=\sqrt{\mathrm{53}}\:{e}^{{itan}^{−\mathrm{1}}…
Question Number 130738 by bemath last updated on 28/Jan/21 $$\int_{\:\mathrm{1}} ^{\:\mathrm{cosh}\:\mathrm{x}} \sqrt{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\:\mathrm{dt}\:=\:?\: \\ $$ Answered by MJS_new last updated on 28/Jan/21 $$\int\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}{dt}= \\…
Question Number 65203 by arcana last updated on 26/Jul/19 $$\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 65202 by arcana last updated on 26/Jul/19 $$\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{cos}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:\theta+{a}^{\mathrm{2}} }{d}\theta,\:{a}^{\mathrm{2}} <\mathrm{1} \\ $$$$\mathrm{answer}? \\ $$ Answered by Tanmay chaudhury last updated on…
Question Number 65200 by rajesh4661kumar@gmail.com last updated on 26/Jul/19 Answered by ajfour last updated on 26/Jul/19 $${I}=\int\sqrt{\mathrm{tan}\:{x}}{dx} \\ $$$${let}\:\mathrm{tan}\:{x}={t}^{\mathrm{2}} \:\Rightarrow\:{dx}=\frac{\mathrm{2}{tdt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$\Rightarrow\:{I}=\int\frac{\mathrm{2}{t}^{\mathrm{2}} {dt}}{\mathrm{1}+{t}^{\mathrm{4}} }…