Question Number 159648 by cortano last updated on 19/Nov/21 $$\:\:{y}\:=\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{2}{x}\right) \\ $$$$\:\:{y}^{\left({n}\right)} \:=?\: \\ $$ Answered by FongXD last updated on 19/Nov/21 $$\bullet\:\mathrm{y}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\mathrm{cos4x}\right)=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos4x} \\…
Question Number 28529 by abdo imad last updated on 26/Jan/18 $${solve}\:{the}\:{d}.{e}.\:\left({x}^{\mathrm{2}} −\mathrm{1}\right){y}^{'} +{xy}=\:{x}^{\mathrm{2}} −{e}^{{x}} \:. \\ $$ Commented by abdo imad last updated on 28/Jan/18…
Question Number 159592 by mnjuly1970 last updated on 19/Nov/21 $$ \\ $$$$\:\:\:{calculate}: \\ $$$$ \\ $$$$\:\:\:\mathcal{I}\::=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{arctan}\left({x}\right)}{{x}}\right)^{\mathrm{3}} {dx}=? \\ $$$$ \\ $$ Answered by…
Question Number 159540 by mnjuly1970 last updated on 18/Nov/21 $$ \\ $$$$\:\: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left(\sqrt{{x}}\:\right).{sin}\left(\frac{\pi}{\mathrm{3}}\:+\sqrt{{x}}\:\right).{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}+\sqrt{{x}}\:\right).{ln}\left(\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }\:\right)}{{x}}{dx}\overset{?} {=}\:\pi.\left(\gamma\:+\:{ln}\left(\mathrm{3}\right)\:\right)\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−−−\:\:\:{m}.{n} \\…
Question Number 159526 by mnjuly1970 last updated on 18/Nov/21 $$ \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left({x}\right){ln}\left({x}\right)}{{x}}\:{dx}\overset{??} {=}\frac{\pi}{\mathrm{8}}\:\left({ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma\right) \\ $$$$−−−−− \\ $$$$\:\:\:\:\:\:{solution}.. \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}^{\:} } ^{\:\infty} \left\{\frac{\frac{\mathrm{3}}{\mathrm{4}}\:{sin}\left({x}\right)−\frac{\mathrm{1}}{\mathrm{4}}\:{sin}\left(\mathrm{3}{x}\right)}{{x}}\right\}\:{ln}\left({x}\right){dx}…
Question Number 93953 by mhmd last updated on 16/May/20 Answered by Rasheed.Sindhi last updated on 16/May/20 $$\left({x}−\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{2}} \\ $$$$\:{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\mathrm{2}=\mathrm{25} \\ $$$$\:{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}}…
Question Number 159474 by mnjuly1970 last updated on 17/Nov/21 $$ \\ $$$$\:\:\:\:\:\:{nice}\:\:{integral}. \\ $$$$\:\:{prove}\:\:{that} \\ $$$$\underset{\mathrm{0}} {\int}^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\mathrm{2}{x}\right)+\:{tan}^{\:−\mathrm{1}} \left(\frac{{x}}{\mathrm{2}}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx}=\frac{\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\…
Question Number 159465 by mnjuly1970 last updated on 17/Nov/21 $$ \\ $$$$\:\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\xi\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\:\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\mathrm{3}} }\:\right)=? \\ $$$$ \\ $$…
Question Number 159447 by mnjuly1970 last updated on 17/Nov/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:#{calculate}# \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{x}^{\:\frac{{t}}{\mathrm{2}}} −{x}^{\:{t}} }{\mathrm{1}\:−\:{x}}\:{dx}\:{dt}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:−−−{m}.{n}−−− \\ $$ Commented…
Question Number 28341 by Cheyboy last updated on 24/Jan/18 $${Find}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{5}{x}−\mathrm{tan}\:\left(\mathrm{5}{x}\right)}{{x}^{\mathrm{3}} } \\ $$ Commented by Cheyboy last updated on 24/Jan/18 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{5}−\mathrm{sec}^{\mathrm{2}} \left(\mathrm{5}{x}\right).\mathrm{5}}{\mathrm{3}{x}^{\mathrm{2}} }…