Question Number 31085 by abdo imad last updated on 02/Mar/18 $${calculate}\:\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:−\mathrm{2}{x}\leqslant\mathrm{0}} {xdxdy}. \\ $$ Commented by abdo imad last updated on 11/Mar/18 $${let}\:{use}\:{the}\:{olar}\:{coordinates}\:\:{x}={rcos}\theta\:{and}\:{y}={rsin}\theta…
Question Number 162151 by Tawa11 last updated on 27/Dec/21 Answered by mr W last updated on 27/Dec/21 $${y}={W}\left(\frac{{x}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\frac{{x}}{\mathrm{2}}={ye}^{{y}} \:\Rightarrow{x}=\mathrm{2}{ye}^{{y}} \\ $$$$\frac{{dx}}{{dy}}=\mathrm{2}{e}^{{y}} +\mathrm{2}{ye}^{{y}} =\mathrm{0}\:\Rightarrow{y}=−\mathrm{1}…
Question Number 162138 by LEKOUMA last updated on 27/Dec/21 Answered by Ar Brandon last updated on 27/Dec/21 $$\mathscr{L}=\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\sqrt{{x}}}{\mathrm{1}−\mathrm{ln}\left({e}−{x}\right)} \\ $$$$\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\sqrt{{x}}}{\mathrm{1}−\mathrm{ln}{e}−\mathrm{ln}\left(\mathrm{1}−\frac{{x}}{{e}}\right)} \\…
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Question Number 31017 by 78987 last updated on 02/Mar/18 $${solve}\:\sqrt{\mathrm{1}+{tan}^{\mathrm{2}} {x}/\mathrm{1}+{cot}^{\mathrm{2}} {x}=\:\:\:{tanx}} \\ $$ Answered by iv@0uja last updated on 02/Mar/18 $$\mathrm{1}+\mathrm{tan}^{\mathrm{2}} {x}=\frac{\mathrm{cos}^{\mathrm{2}} {x}}{\mathrm{cos}^{\mathrm{2}} {x}}+\frac{\mathrm{sin}^{\mathrm{2}}…
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Question Number 30771 by abdo imad last updated on 25/Feb/18 $${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} }\:\:\:\:\left({n}\in{N}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{fromR}\:/\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$$$\frac{\mathrm{1}}{{cosx}}=\frac{{acosx}}{\mathrm{1}−{sinx}}\:+\frac{{bcosx}}{\mathrm{1}+{sinx}}\:\:.{find}\:\:{I}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{verify}\:{the}\:{relation} \\ $$$$\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} {x}}=\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}\:+\frac{{sinx}\:{sinx}}{{cos}^{\mathrm{2}{n}+\mathrm{3}}…
Question Number 96290 by 175 last updated on 31/May/20 $${If}\::\:\mathrm{tan}\left({x}\:+{iy}\right)\:=\:{a}\:+\:{bi}\: \\ $$$${then}\:{find}\:{a},{b} \\ $$ Commented by Tony Lin last updated on 31/May/20 $${tan}\left({x}+{iy}\right) \\ $$$$=\frac{{tanx}+{taniy}}{\mathrm{1}−{tanxtaniy}}…
Question Number 161723 by LEKOUMA last updated on 21/Dec/21 $${Calculate} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}+{x}.\mathrm{2}^{{x}} }{\mathrm{1}+{x}.\mathrm{3}^{{x}} }\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\mathrm{2}{e}^{\frac{{x}}{{x}+\mathrm{1}}} −\mathrm{1}\right]^{\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}}} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{x}} −{a}^{{a}}…
Question Number 161644 by talminator2856791 last updated on 20/Dec/21 $$\: \\ $$$$\:\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{single}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{symbol}\:\: \\ $$$$\:\mathrm{the}\:\mathrm{closest}\:\mathrm{to}\:\mathrm{such}\:\mathrm{a}\:\mathrm{symbol}\:\mathrm{is}\:\ll\:\mathrm{or}\:\gg\:\: \\ $$$$\:\mathrm{when}\:\mathrm{is}\:\mathrm{tinku}\:\mathrm{tara}\:\mathrm{going}\:\mathrm{to}\:\mathrm{add}\:\mathrm{the}\:\mathrm{single}\:\mathrm{arrow}\:\: \\ $$$$\:\mathrm{its}\:\mathrm{a}\:\mathrm{very}\:\mathrm{common}\:\mathrm{symbol}\:\mathrm{and}\:\mathrm{should}\:\mathrm{be}\:\mathrm{on}\:\mathrm{this}\:\mathrm{keyboard}\:\: \\ $$$$\: \\ $$$$\: \\ $$ Commented…