Question Number 215994 by Frix last updated on 25/Jan/25 $$\mathrm{Solve}\:\mathrm{for}\:{z}\in\mathbb{C}:\:\:\:\:\:\mid{z}^{{z}} \mid=\mathrm{1} \\ $$ Answered by mr W last updated on 25/Jan/25 $${say}\:{z}={re}^{{i}\theta} ={r}\left(\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta\right) \\ $$$${with}\:\mathrm{0}\leqslant\theta<\mathrm{2}\pi,\:{r}>\mathrm{0}…
Question Number 215995 by universe last updated on 25/Jan/25 $$\int\int\underset{{D}} {\int}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }\:\mathrm{dv}\:=\:? \\ $$$$\mathrm{D}\:=\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} <{z} \\ $$ Answered by mr W…
Question Number 216016 by Tawa11 last updated on 25/Jan/25 Answered by som(math1967) last updated on 25/Jan/25 $${current}\:\mathrm{20}\Omega\:{when}\:{both}\:{switch} \\ $$$${oppen}=\frac{\mathrm{6}}{\mathrm{80}}=\frac{\mathrm{3}}{\mathrm{40}}{amp} \\ $$$${current}\:{flow}\:{cct}\:{when}\:{s}_{\mathrm{1}} \:{s}_{\mathrm{2}} \:{both} \\ $$$${close}\:\mathrm{6}\boldsymbol{\div}\left(\mathrm{50}+\frac{\mathrm{20}×{R}}{\mathrm{20}+{R}}\right)…
Question Number 215981 by ajfour last updated on 24/Jan/25 Commented by ajfour last updated on 24/Jan/25 $${Upper}\:{string}\:{end}\:{to}\:{top}\:{corner}\:{angle}\:{be}\:\boldsymbol{{c}}. \\ $$ Commented by Tawa11 last updated on…
Question Number 215992 by MATHEMATICSAM last updated on 24/Jan/25 $$\left(\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\right)^{{x}} \:+\:\left(\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}}\right)^{{x}} \:=\:\mathrm{10}.\:\mathrm{Solve} \\ $$$$\mathrm{for}\:{x}. \\ $$ Answered by Rasheed.Sindhi last updated on 24/Jan/25 $$\left(\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\right)^{{x}} \:+\:\left(\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}}\right)^{{x}}…
Question Number 215991 by F_Physics last updated on 24/Jan/25 $$\underset{{x}=\mathrm{1}} {\overset{{y}} {\sum}}{x}^{{n}} =\mathrm{0} \\ $$$${n}=? \\ $$ Commented by mr W last updated on 25/Jan/25…
Question Number 215963 by RoseAli last updated on 23/Jan/25 Answered by Frix last updated on 23/Jan/25 $$\int\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)^{{n}} \left(\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{x}\right){dx}= \\ $$$$=\int\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)^{{n}} \left(\mathrm{sin}^{\mathrm{2}} \:{x}\:−\mathrm{cos}^{\mathrm{2}} \:{x}\right){dx}= \\…
Question Number 215979 by alcohol last updated on 23/Jan/25 $${u}_{{n}} \:=\:\underset{{k}={n}+\mathrm{1}} {\overset{\mathrm{2}{n}} {\sum}}\frac{\mathrm{1}}{{k}}\:{and}\:{v}_{{n}} \:=\:\underset{{k}={n}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{{k}} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:{and}\:{v}_{{n}} \:{are}\:{adjacent} \\ $$$${use}\:{ln}\left({x}+\mathrm{1}\right)\:\leqslant\:{x}\:{and}\:{x}\leqslant−{ln}\left(\mathrm{1}−{x}\right)\:{and} \\ $$$$\bullet\:{show}\:{that}\:{u}_{{n}} \:\leqslant\:\underset{{k}={n}+\mathrm{1}}…
Question Number 215973 by MATHEMATICSAM last updated on 23/Jan/25 $$\left({a}\:+\:{b}\right)\:\propto\:{c}\:\mathrm{and}\:\left({b}\:+\:{c}\right)\:\propto\:{a}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \\ $$ Answered by Rasheed.Sindhi last updated on 24/Jan/25 $$\left({a}\:+\:{b}\right)\:\propto\:{c}\:\mathrm{and}\:\left({b}\:+\:{c}\right)\:\propto\:{a}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \\…
Question Number 215974 by ajfour last updated on 23/Jan/25 $$\int\sqrt{\left(\mathrm{2sin}^{−\mathrm{1}} {x}−{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right)^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$ Answered by MathematicalUser2357 last updated on 26/Jan/25 $$\mathrm{No}\:\mathrm{closed}\:\mathrm{forms}.\:\frac{{x}\mid{x}\mid}{\mathrm{2}}+\frac{{x}^{\mathrm{3}} \mid{x}\mid}{\mathrm{3}}−\frac{\mathrm{2}{x}^{\mathrm{5}}…