Question Number 8996 by Basant007 last updated on 11/Nov/16 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x} \\ $$ Commented by FilupSmith last updated on 13/Nov/16 $${y}=\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right) \\ $$$${u}=\mathrm{sin}\left(\mathrm{2}{x}\right)\:\Rightarrow\:{du}=\mathrm{2cos}\left(\mathrm{2}{x}\right){dx} \\…
Question Number 140055 by mnjuly1970 last updated on 03/May/21 $$\:\:\:\:\:\: \\ $$$$\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\:\:\Omega:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}−{e}^{−{x}} }{\mathrm{1}+{e}^{\mathrm{2}{x}} }\:.\frac{{dx}}{{x}}\:={ln}\left(\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{4}\sqrt{\mathrm{2}\pi}}\:\right) \\ $$$$\:\Theta:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}{n}}\right)^{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} } \overset{??}…
Question Number 140050 by mnjuly1970 last updated on 03/May/21 $$\:\:\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\:\:\:\phi\::={lim}_{{n}\rightarrow\infty} \frac{{n}}{\:\sqrt{\mathrm{2}{k}}}\:.\sqrt{\mathrm{1}−{cos}^{{k}} \left(\frac{\mathrm{2}\pi}{{n}}\right)}\:=\pi \\ $$$$\:\:\:\:\:\:\:\:……………. \\ $$$$ \\ $$ Answered by Kamel last updated…
Question Number 139949 by mnjuly1970 last updated on 02/May/21 $$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:………{nice}\:….\:\ast\:….\ast\:….\:\ast\:….{calculus}\left({I}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\:\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left(\sqrt[{\mathrm{3}}]{{cos}\left(\sqrt{{x}}\:\right.}\:\right)^{{cot}\left({x}\right)} =?? \\ $$$$\:\:\:\:\:\:\:\:{solution}……… \\ $$$$\:\:\:\:\:\:\:{sin}\left({x}\right)\:\approx\:{x}\:\:\:\:\:\:\:\:\left({x}\:\rightarrow\:\mathrm{0}\:\right) \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}:=\:\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left\{\left({cos}\left(\sqrt{{x}}\:\right)\right)^{\frac{\mathrm{1}}{\mathrm{3}}}…
Question Number 139878 by bramlexs22 last updated on 02/May/21 $$\:\:\:\:\:\:\:\mathrm{I}\left(\alpha\right)\:=\:\underset{\alpha} {\overset{\alpha^{\mathrm{2}} } {\int}}\:\frac{\mathrm{sin}\:\alpha\mathrm{x}}{\mathrm{x}}\:\mathrm{dx}\: \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{dI}\left(\alpha\right)}{\mathrm{d}\alpha}\:=? \\ $$ Answered by EDWIN88 last updated on 02/May/21 $$\:\:\:\:\:\frac{\mathrm{dI}}{\mathrm{d}\alpha}\:=\:\underset{\alpha}…
Question Number 139851 by mnjuly1970 last updated on 01/May/21 $$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:…… \\ $$$$\:\:\:\:\:\:\Phi=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)}{\:\sqrt{{x}}}{dx}=\mathrm{4}\:{ln}\left(\frac{{e}}{\mathrm{2}}\right)\:…\checkmark \\ $$ Answered by mindispower last updated on 01/May/21…
Question Number 74235 by aliesam last updated on 20/Nov/19 Answered by Joel578 last updated on 20/Nov/19 $$\mid{x}\mid\:=\:\sqrt{{x}^{\mathrm{2}} } \\ $$$${f}\left({x}\right)\:=\:\sqrt{\left({x}^{\mathrm{2}} \:−\:\mathrm{3}\right)^{\mathrm{2}} } \\ $$$${f}\:'\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\left({x}^{\mathrm{2}} \:−\:\mathrm{3}\right)^{\mathrm{2}}…
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Question Number 8488 by Basant007 last updated on 12/Oct/16 $${find}\:{y}_{{n}} \:{if}\:{y}=\mathrm{cos}\:\mathrm{2}{x} \\ $$ Commented by FilupSmith last updated on 13/Oct/16 $${y}_{{n}} =\frac{{d}^{{n}} {y}}{{dx}^{{n}} } \\…
Question Number 139555 by snipers237 last updated on 28/Apr/21 $$\:\:\underset{{i}=\mathrm{0}} {\overset{\mathrm{5}} {\prod}}\left(\mathrm{1}−{cotan}\left(\mathrm{20}+{i}\right)\right)\:\:\overset{?} {=}\:\mathrm{8} \\ $$ Answered by mr W last updated on 28/Apr/21 $$\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{tan}\:\mathrm{20}°}\right)\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{tan}\:\mathrm{25}°}\right) \\…