Question Number 61232 by maxmathsup by imad last updated on 30/May/19 $${let}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\left[{nx}\right]−\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\:\:{with}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty}…
Question Number 126766 by mnjuly1970 last updated on 24/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{calculus}\:\:\left({I}\right)… \\ $$$$\:\:\:\:{please}\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{6}} }\:\right){dx}=? \\ $$$$ \\ $$ Answered by Ar Brandon…
Question Number 126764 by BHOOPENDRA last updated on 24/Dec/20 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 61229 by maxmathsup by imad last updated on 30/May/19 $${let}\:{f}_{{n}} \left({a}\right)\:=\int_{\mathrm{0}} ^{{a}} \:{x}^{{n}} \sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}_{{n}} \left({a}\right)\:={f}^{'} \left({a}\right)\:\:\:{give}\:{g}_{{n}} \left({a}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{give}\:{its}…
Question Number 126765 by mnjuly1970 last updated on 24/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{elementary}\:\:{mathematics}… \\ $$$$\:\:\:{if}\:\:\:\:\:\mathrm{13}\:\mid\mathrm{9}^{\mathrm{51}} +{k}+\mathrm{1}\:\:\:,\:{k}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{then}\:\:\:{k}_{\left({min}\right)} \:=? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$ Answered by floor(10²Eta[1])…
Question Number 192297 by Mastermind last updated on 14/May/23 Answered by witcher3 last updated on 14/May/23 $$\mathrm{let}\:\mathrm{a}=\mathrm{supA},\mathrm{b}=\mathrm{SupB}\Rightarrow\forall\left(\mathrm{x}\in\mathrm{Aety}\in\mathrm{B}\right) \\ $$$$\mathrm{x}+\mathrm{y}\leqslant\mathrm{a}+\mathrm{b}\Rightarrow\mathrm{sup}\left(\mathrm{A}+\mathrm{B}\right)\leqslant\mathrm{a}+\mathrm{b} \\ $$$$\mathrm{let}\:\mathrm{M}=\mathrm{sup}\left(\mathrm{A}+\mathrm{B}\right) \\ $$$$\forall\epsilon>\mathrm{0}\:\exists\mathrm{x}\in\mathrm{A},\mathrm{y}\in\mathrm{B}\:\mathrm{such}\:\mathrm{a}−\epsilon<\mathrm{x}\leqslant\mathrm{a} \\ $$$$\mathrm{b}−\epsilon<\mathrm{y}\leqslant\mathrm{b}…
Question Number 192299 by Mastermind last updated on 14/May/23 Answered by witcher3 last updated on 14/May/23 $$\mathrm{R}\subseteq\mathrm{S}\Rightarrow\mathrm{x}=\mathrm{Sup}\left(\mathrm{R}\right)\leqslant\mathrm{sup}\left(\mathrm{S}\right)=\mathrm{y} \\ $$$$\forall\mathrm{r}\in\mathrm{R}\:\:\mathrm{r}\leqslant\mathrm{y},\forall\epsilon>\mathrm{0}\:\exists\mathrm{s}\in\mathrm{S}\:\mathrm{such}\:\mathrm{y}−\epsilon<\mathrm{s}\leqslant\mathrm{y} \\ $$$$\mathrm{by}\:\mathrm{definition}\:\exists\mathrm{r}\in\mathrm{R}\:\mathrm{r}\geqslant\mathrm{s} \\ $$$$\Rightarrow\forall\epsilon>\mathrm{0}\:\mathrm{y}−\epsilon\leqslant\mathrm{r} \\ $$$$\epsilon=\frac{\mathrm{1}}{\mathrm{n}}\Rightarrow\forall\mathrm{n}\in\mathbb{N}\:\:\mathrm{y}−\frac{\mathrm{1}}{\mathrm{n}}<\mathrm{r}\Rightarrow\mathrm{s}\geqslant\mathrm{y}\Rightarrow\mathrm{sup}\left(\mathrm{S}\right)\leqslant\mathrm{Sup}\left(\mathrm{R}\right)…
Question Number 126761 by bemath last updated on 24/Dec/20 Answered by liberty last updated on 24/Dec/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{3}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\mathrm{1}}{\mathrm{8}}{x}^{\mathrm{2}} +{O}\left({x}^{\mathrm{3}} \right)\right)−\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+{O}\left({x}^{\mathrm{5}} \right)\right)^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{3}} \left({x}+\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}}…
Question Number 126758 by bounhome last updated on 24/Dec/20 $${solve}: \\ $$$$\mathrm{1}.{y}'^{\mathrm{3}} +{y}^{\mathrm{2}} −{y}'^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{2}.\mathrm{4}{y}={x}^{\mathrm{2}} +{y}'^{\mathrm{2}} \\ $$ Answered by liberty last updated…
Question Number 192289 by cortano12 last updated on 14/May/23 $$\:\:\:\:\:\:\:\:\underset{−\mathrm{e}} {\overset{\mathrm{e}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}}\:\mathrm{dx}\:=? \\ $$ Answered by mehdee42 last updated on 14/May/23 $${f}\left({x}\right)=\frac{{sinx}}{{secx}^{\mathrm{2}} {x}+\mathrm{1}}\Rightarrow{f}\left(−{x}\right)={f}\left({x}\right) \\…