Question Number 193734 by Mastermind last updated on 18/Jun/23 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$ Answered by witcher3…
Question Number 193612 by Noorzai last updated on 17/Jun/23 Answered by aba last updated on 17/Jun/23 $$\mathrm{x}^{\mathrm{x}^{\mathrm{2}} } =\mathrm{3}\:\Rightarrow\:\mathrm{x}^{\mathrm{2}} \mathrm{lnx}=\mathrm{ln3} \\ $$$$\Rightarrow\mathrm{2lnx}.\mathrm{e}^{\mathrm{2lnx}} =\mathrm{2ln3}\:\Rightarrow\:\mathrm{2lnx}=\mathrm{W}\left(\mathrm{2ln3}\right) \\ $$$$\Rightarrow\:\mathrm{x}=\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{W}\left(\mathrm{2ln3}\right)}…
Question Number 193513 by Mastermind last updated on 15/Jun/23 $$\mathrm{Evaluate}\:\mathrm{I}=\underset{\mathrm{s}} {\int}\int\mathrm{x}^{\mathrm{3}} \mathrm{dydz}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{ydzdx}. \\ $$$$\mathrm{where}\:\mathrm{S}\:\mathrm{is}\:\mathrm{the}\:\mathrm{closed}\:\mathrm{surface}\:\mathrm{consis}− \\ $$$$\mathrm{ting}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cylinder}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} ,\:\mathrm{0}\leqslant\mathrm{z}\leqslant\mathrm{b} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{cylinder}\:\:\mathrm{disks}\:\mathrm{z}=\mathrm{0}\:\mathrm{and}\:\mathrm{z}=\mathrm{b}, \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}}…
Question Number 193323 by fit last updated on 10/Jun/23 Answered by MM42 last updated on 10/Jun/23 $$\left.\mathrm{9}\right)\:{n}\left({s}\right)=\mathrm{80}\:\&\:{n}\left({m}\right)=\mathrm{40}\:\&\:{n}\left({e}\right)=\mathrm{65} \\ $$$${n}\left({s}\right)={n}\left({m}\right)+{n}\left({e}\right)−{n}\left({m}\cap{e}\right) \\ $$$$\mathrm{80}=\mathrm{40}+\mathrm{65}−{n}\left({m}\cap{e}\right)\Rightarrow{n}\left({m}\cap{e}\right)=\mathrm{15} \\ $$$$\left.\mathrm{10}\right)\frac{\mathrm{15}!}{\mathrm{3}!\mathrm{2}!\mathrm{2}!\mathrm{2}!\mathrm{2}!\mathrm{2}!} \\ $$$$\left.\mathrm{11}\right)\mathrm{10}×\mathrm{9}×\mathrm{8}×\mathrm{7}…
Question Number 193137 by Mastermind last updated on 04/Jun/23 $$\left.\mathrm{1}\right)\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mid\mathrm{a}+\mathrm{b}+\mathrm{c}\mid\geqslant\mid\mathrm{a}\mid−\mid\mathrm{b}\mid−\mid\mathrm{c}\mid \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Find}\:\mathrm{all}\:\mathrm{x}\in\mathbb{R}\:\mathrm{that}\:\mathrm{satify}\:\mathrm{the}\:\mathrm{follow}− \\ $$$$\mathrm{ing}\:\mathrm{inequalities}\: \\ $$$$\left.\mathrm{i}\right)\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{4}\mid<\mathrm{5} \\ $$$$\left.\mathrm{ii}\right)\:\mid\mathrm{x}\mid+\mid\mathrm{x}+\mathrm{2}\mid<\mathrm{5} \\ $$$$…
Question Number 193138 by Mastermind last updated on 04/Jun/23 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{a},\mathrm{b}\in\mathbb{R} \\ $$$$\left.\mathrm{i}\right)\:\mathrm{ab}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{ii}\right)\:\left(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}\right)^{\mathrm{2}} \leqslant\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{iii}\right)\:\sqrt{\mathrm{ab}}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{a}+\mathrm{b}\right),\:\mathrm{for}\:\mathrm{a},\mathrm{b}\geqslant\mathrm{0}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{have}\:\mathrm{square}\:\mathrm{roots}. \\ $$$$…
Question Number 193139 by Mastermind last updated on 04/Jun/23 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{inequalities}: \\ $$$$\mid\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mid>\mathrm{1} \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$ Answered by Skabetix last updated on 04/Jun/23…
Question Number 193116 by Mastermind last updated on 04/Jun/23 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\in\:\mathbb{R}\:\mathrm{with} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\geqslant\:\mathrm{0}\: \\ $$$$\left.\mathrm{1}\right)\:\sqrt{\mathrm{ab}}\sqrt{\mathrm{cd}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{2}\right)\:\left(\mathrm{abcd}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right) \\ $$$$ \\ $$$$\mathrm{Help}!…
Question Number 193109 by Mastermind last updated on 04/Jun/23 Answered by math1234 last updated on 04/Jun/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 193154 by wizzy1 last updated on 04/Jun/23 Commented bywizzy1 last updated on 04/Jun/23 Terms of Service Privacy Policy Contact: info@tinkutara.com