Question Number 172311 by mathocean1 last updated on 25/Jun/22 $${Using}\:{Riemann}'{s}\:{sum},\:{calculate}: \\ $$$${lim}\:{b}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\mathrm{2}\left(\frac{{kn}}{{n}}\right) \\ $$ Commented by JDamian last updated on 25/Jun/22 $$!!!\:\:\frac{{k}\cancel{{n}}}{\cancel{{n}}}={k}…
Question Number 172304 by mathocean1 last updated on 25/Jun/22 $${study}\:{the}\:{convergence}\:{of}: \\ $$$${I}\left(\alpha\right)=\int_{\mathrm{1}} ^{\alpha} \frac{\mathrm{1}}{{t}^{\alpha} \left(\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}\right)}{dt},\:{in}\:{function}\: \\ $$$${of}\:{real}\:\alpha. \\ $$$${Calculate}\:{I}\left(\alpha\right)\:{for}\:\alpha=\mathrm{1};\mathrm{2};\mathrm{4}.\: \\ $$ Terms of Service…
Question Number 172307 by mathocean1 last updated on 25/Jun/22 $${Calculate}\: \\ $$$$\underset{{n}\rightarrow+\infty} {{lim}A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx} \\ $$ Answered by aleks041103 last updated on…
Question Number 172306 by mathocean1 last updated on 25/Jun/22 $${show}\:{that}\:{J}=\int_{\mathrm{0}} ^{+\infty} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dt}\:{with}\:{a}>\mathrm{0} \\ $$$${is}\:{convergent} \\ $$ Answered by aleks041103 last updated on 25/Jun/22…
Question Number 106743 by mohammad17 last updated on 06/Aug/20 $$\int\sqrt{{secy}}{dy} \\ $$ Answered by Sarah85 last updated on 07/Aug/20 $$\int\sqrt{\mathrm{sec}\:{y}}{dy}=\int\frac{{dy}}{\:\sqrt{\mathrm{cos}\:{y}}}=\int\frac{{dy}}{\:\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:\frac{{y}}{\mathrm{2}}}} \\ $$$$\mathrm{let}\:{t}=\frac{{y}}{\mathrm{2}} \\ $$$$\mathrm{2}\int\frac{{dt}}{\:\sqrt{\mathrm{1}−\mathrm{2sin}^{\mathrm{2}}…
Question Number 41198 by mondodotto@gmail.com last updated on 03/Aug/18 $$\mathrm{evaluate}\:\boldsymbol{\mathrm{ln}}\left(−\mathrm{1}\right) \\ $$ Commented by math khazana by abdo last updated on 03/Aug/18 $${we}\:\:{have}\:\:−\mathrm{1}\:={e}^{{i}\left(\mathrm{2}{k}+\mathrm{1}\right)\pi} \:\:\:\:\:\forall{k}\in{Z}\:\Rightarrow \\…
Question Number 106726 by ZiYangLee last updated on 06/Aug/20 $$\mathrm{Prove}\:\mathrm{sin5}\theta=\mathrm{16sin}^{\mathrm{5}} \theta−\mathrm{20sin}^{\mathrm{3}} \theta+\mathrm{5sin}\theta \\ $$$$\mathrm{Hence},\:\mathrm{show}\:\mathrm{that}\:\mathrm{sin}\:\mathrm{6}°\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{irrational}\:\mathrm{number}.\: \\ $$ Answered by Ar Brandon last updated on…
Question Number 106690 by M±th+et+s last updated on 06/Aug/20 $${this}\:{question}\:{was}\:{repeatd}\:{six}\:{times} \\ $$$${in}\:{a}\:{various}\:{exams}\:{between}\:\mathrm{1971}\:{to}\:\mathrm{2001}. \\ $$$${if}\:{C}_{\mathrm{0}} ,{C}_{\mathrm{1}} ,{C}_{\mathrm{2}} …….,{C}_{{n}} \:{are}\:{the}\:{coefficients} \\ $$$${in}\:{the}\:{expansion}\:{of}\:\left(\mathrm{1}+{x}\right)^{{n}} \:{then} \\ $$$${c}_{\mathrm{0}} +\mathrm{2}{C}_{\mathrm{1}} +\mathrm{3}{C}_{\mathrm{2}}…
Question Number 106677 by mohammad17 last updated on 06/Aug/20 Answered by Dwaipayan Shikari last updated on 06/Aug/20 $$\left.\mathrm{5}\right)\int_{\mathrm{1}} ^{\mathrm{4}} \left(\mathrm{1}−{u}\right)\sqrt{{u}}{du} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{4}} \sqrt{{u}}−{u}^{\frac{\mathrm{3}}{\mathrm{2}}} {du}…
Question Number 106662 by mathdave last updated on 06/Aug/20 Answered by Ar Brandon last updated on 06/Aug/20 $$\mathrm{Let}\:\mathrm{A}_{\mathrm{n}} =\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{ln}\left(\mathrm{x}!\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \:,\:\mathrm{x}\in\mathbb{N} \\ $$$$\mathrm{lnx}!=\mathrm{ln}\left(\left(\mathrm{x}\right)\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{2}\right)…\left(\mathrm{x}−\left(\mathrm{x}−\mathrm{1}\right)\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:=\mathrm{ln}\underset{\mathrm{k}=\mathrm{0}}…