Question Number 63507 by mathmax by abdo last updated on 05/Jul/19 $${let}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\right){dx}\:\:\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{nature}\:{of}\:\:\Sigma\:{U}_{{n}} \\ $$ Answered…
Question Number 63499 by Rio Michael last updated on 04/Jul/19 $${given}\:{that}\:\:\:{a}\mid{b},\:{show}\:{that}\:−{a}\mid{b}. \\ $$ Answered by MJS last updated on 04/Jul/19 $${a}\mid{b}\:\Rightarrow\:\frac{{b}}{{a}}={c}\:\mathrm{with}\:{c}\in\mathbb{Z} \\ $$$$\Rightarrow\:−\frac{{b}}{{a}}=\left(−\mathrm{1}\right)\frac{{b}}{{a}}=\left(−\mathrm{1}\right){c}=−{c} \\ $$$${c}\in\mathbb{Z}\:\Leftrightarrow\:−{c}\in\mathbb{Z}…
Question Number 129018 by BHOOPENDRA last updated on 12/Jan/21 $$\frac{\left(\sqrt{{s}}−\mathrm{1}\right)^{\mathrm{2}} }{{s}^{\mathrm{2}} }\:{find}\:{the}\:{inverse}\:{laplace}\:{transformtion} \\ $$ Answered by Dwaipayan Shikari last updated on 12/Jan/21 $$\mathscr{L}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{s}}−\frac{\mathrm{2}}{\:{s}^{\frac{\mathrm{3}}{\mathrm{2}}} }+\frac{\mathrm{1}}{{s}^{\mathrm{2}}…
Question Number 129010 by BHOOPENDRA last updated on 12/Jan/21 Commented by BHOOPENDRA last updated on 12/Jan/21 $${find}\:{the}\:{inverse}\:{laplace}\:{transformation}? \\ $$ Answered by Dwaipayan Shikari last updated…
Question Number 63447 by minh2001 last updated on 04/Jul/19 $${How}\:{to}\:{calculate}\:\lceil\left({n}\right)\: \\ $$$${using}\:{gamma}\:{function} \\ $$$$\forall{n}\in{R} \\ $$ Commented by minh2001 last updated on 04/Jul/19 $${I}\:{can}'{t}\:{remember}\:{it}\:{until}\: \\…
Question Number 63428 by Rio Michael last updated on 04/Jul/19 $${It}\:{is}\:{given}\:{that}\:{S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}^{\mathrm{2}\:} −\mathrm{3}{r}−\mathrm{1}\right).\:{Use}\:{the}\:{the}\:{formulae} \\ $$$${of}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}\:\:} {and}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\:\:{to}\:{show}\:{that}\:{S}_{{n}} ={n}^{\mathrm{3}} . \\…
Question Number 63424 by Rio Michael last updated on 04/Jul/19 $${A}\:{colony}\:{of}\:{bacteria}\:{if}\:{left}\:{undisturbed}\:{will}\:{grow}\:{at}\:{a}\:{rate} \\ $$$${proportional}\:{to}\:{the}\:{number}\:{of}\:{bacteria},\:{P}\:{present}\:{at}\:{time},{t}. \\ $$$${However},{a}\:{toxic}\:{substance}\:{is}\:{being}\:{added}\:{slowly}\:{such}\:{that} \\ $$$${at}\:{time}\:{t},\:{the}\:{bacteria}\:{also}\:{die}\:{at}\:{the}\:{rate}\:\mu{Pt}\:{where}\:\mu\:{is} \\ $$$${a}\:{positive}\:{constant}. \\ $$$$\left({a}\right)\:\:{Show}\:{that}\:{at}\:{time}\:{t}\:{the}\:{rate}\:{of}\:{growth}\:{of}\:{the}\:{bacteria}\:{in} \\ $$$${the}\:{colony}\:{is}\:{governed}\:{by}\:{the}\:{differential}\:{equation} \\ $$$$\:\frac{{dP}}{{dt}}=\:\left({k}−\mu{t}\right){p}\:{where}\:{k}\:{is}\:{apositive}\:{constant}.…
Question Number 63425 by Rio Michael last updated on 04/Jul/19 $${The}\:{probability}\:{that}\:{a}\:{vaccinated}\:{person}\left({V}\right)\:{contracts}\:{a}\:{disease} \\ $$$${is}\:\frac{\mathrm{1}}{\mathrm{20}}.\:{For}\:{a}\:{person}\:{vaccinated}\left({V}\:'\right)\:,\:{the}\:{probability}\:{of}\:{contracting} \\ $$$${a}\:{disease}\:{is}\:\frac{\mathrm{5}}{\mathrm{6}}.\:{In}\:{a}\:{certain}\:{town}\:\mathrm{90\%}{of}\:{thepopulation}\:{has} \\ $$$${been}\:{vaccinated}\:{against}\:{a}\:{disease}.\:{A}\:{person}\:{is}\:{selected}\:{at} \\ $$$${random}\:{from}\:{the}\:{town},{find}\:{the}\:{probability}\:{that}: \\ $$$$\left({a}\right)\:{he}\:{has}\:{the}\:{disease}, \\ $$$$\left({b}\right)\:{he}\:{is}\:{vaccinated}\:{or}\:{he}\:{has}\:{the}\:{disease}. \\ $$$${sir}\:{Forkum}\:{Michael}…
Question Number 128931 by Dwaipayan Shikari last updated on 11/Jan/21 $${Approximate} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\sqrt{{n}}}{{n}^{\mathrm{2}} +\mathrm{1}} \\ $$ Commented by Dwaipayan Shikari last updated on…
Question Number 128893 by BHOOPENDRA last updated on 11/Jan/21 Answered by Dwaipayan Shikari last updated on 11/Jan/21 $$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {e}^{−\mathrm{3}{t}} {t}^{\mathrm{2}} {e}^{\mathrm{2}{t}} −{e}^{−\mathrm{3}{t}−\mathrm{2}{t}} {t}^{\mathrm{2}} {dt}\:\:\:\:\:\:\:\:\:\:{sinh}\mathrm{2}{t}=\frac{{e}^{\mathrm{2}{t}}…