Question Number 152947 by mnjuly1970 last updated on 03/Sep/21 $$ \\ $$$$\:\:\:{prove}\::: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{−\infty} ^{\:\infty} \:\frac{\:{e}^{\:−\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }} }{{x}^{\:\mathrm{4}} }\:{dx}\:\overset{?} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\Gamma\:\left(\frac{\mathrm{1}}{\mathrm{2}}\:\right) \\ $$$$ \\ $$ Answered…
Question Number 87378 by jagoll last updated on 04/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{3x}−\mathrm{3tan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} } \\ $$ Commented by john santu last updated on 04/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{3x}+\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{3x}\right)^{\mathrm{3}} +\mathrm{o}\left(\left(\mathrm{3x}\right)^{\mathrm{3}}…
Question Number 152904 by DELETED last updated on 03/Sep/21 Answered by DELETED last updated on 03/Sep/21 $$\left.\mathrm{1}\right).\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{4cos}\:\mathrm{x}+\mathrm{5sin}\:\mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:\:=−\mathrm{4}\:\mathrm{sin}\:\mathrm{x}+\mathrm{5}\:\mathrm{cos}\:× \\ $$$$\left.\mathrm{2}\right).\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3}\:\mathrm{sin}\:\mathrm{2x}\:−\:\mathrm{5}\:\mathrm{cos}\:\mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:=\mathrm{3}×\mathrm{2}\:\mathrm{cos}\:\mathrm{2x}\:+\mathrm{5}\:\mathrm{sin}\:\mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{6}\:\mathrm{cos}\:\mathrm{2x}\:+\:\mathrm{5}\:\mathrm{sin}\:\mathrm{x}//…
Question Number 152797 by mnjuly1970 last updated on 01/Sep/21 $$ \\ $$$$\:\:\:{nice}..{mathematics}… \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}… \\ $$$$\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{cos}\:\left({x}\:\right)}{{cosh}\:\left({x}\:\right)}\:{dx}=\frac{\pi}{\:{cosh}\:\left(\frac{\pi}{\mathrm{2}}\:\right)}\:…….\blacksquare\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:{prepared}\:::\:\:{m}.{n} \\ $$$$ \\…
Question Number 152795 by mnjuly1970 last updated on 01/Sep/21 $$ \\ $$$$\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}\:\right)}{{x}}\left(\frac{{a}^{\:\mathrm{2}} +{cos}^{\:\mathrm{2}} \left({x}\right)}{{b}^{\:\mathrm{2}} +\:{cos}^{\:\mathrm{2}} \left({x}\:\right)}\right){dx}=? \\ $$$$ \\ $$ Answered by Olaf_Thorendsen…
Question Number 152761 by mnjuly1970 last updated on 01/Sep/21 $$ \\ $$$$\:{prove}\:{that}: \\ $$$$ \\ $$$$\:\:\mathrm{S}\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\:\mathrm{1}}{{sinh}\:\left(\mathrm{2}^{\:{n}} .{x}\right)}\right)\:\overset{?} {=}\frac{\:\mathrm{2}}{{e}^{\:\mathrm{2}{x}} −\mathrm{1}} \\ $$$$\:{m}.{n}… \\ $$…
Question Number 152653 by mnjuly1970 last updated on 30/Aug/21 $$ \\ $$$$\:\:\Omega\::=\int_{\mathrm{0}} ^{\:\infty} \frac{\:{e}^{\:−{x}^{\:\mathrm{3}} } .\:{sin}\:\left({x}^{\:\mathrm{3}} \:\right)}{{x}}{dx}=\:\frac{\zeta\:\left(\mathrm{2}\:\right)}{\mathrm{2}} \\ $$$$\:{m}.{n}… \\ $$$$ \\ $$ Answered by…
Question Number 87116 by lémùst last updated on 02/Apr/20 $$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+{x}^{\mathrm{2}} {y}=\mathrm{0} \\ $$ Answered by redmiiuser last updated on 03/Apr/20 $$\frac{{y}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{4}} .{y}}{\mathrm{12}}=\mathrm{0}…
Question Number 152631 by mnjuly1970 last updated on 30/Aug/21 Answered by Olaf_Thorendsen last updated on 30/Aug/21 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} +{x}^{\mathrm{4}} \right)}{{x}^{\mathrm{2}} }\:{dx} \\ $$$$\mathrm{I}\:=\:\left[−\frac{\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} +{x}^{\mathrm{4}}…
Question Number 152601 by mnjuly1970 last updated on 30/Aug/21 $$ \\ $$$$\:\:\:{solve}…. \\ $$$$\:\:{lim}_{\:{n}\rightarrow\infty} \left\{\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}\:−\frac{{k}}{{n}}+\frac{{k}^{\:\mathrm{2}} }{{n}^{\:\mathrm{2}} }\:\right)^{\:\frac{\mathrm{1}}{{n}}} \right\}=? \\ $$$$\:\:{m}.{n}… \\ $$$$ \\…