Question Number 143109 by mnjuly1970 last updated on 10/Jun/21 Commented by mnjuly1970 last updated on 10/Jun/21 $$\:\:\:\:\:{prove}\:\:\mathrm{I}_{\mathrm{1}} =−\frac{\pi^{\mathrm{2}} }{\mathrm{192}}\:\:…..? \\ $$ Answered by qaz last…
Question Number 143086 by mnjuly1970 last updated on 09/Jun/21 $$ \\ $$$$\:\:{Evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{{ln}\left({tan}\left({x}\right)\right).{sin}^{\pi^{{e}} } \left(\mathrm{2}{x}\right)}{\left({sin}^{\pi^{{e}} } \left({x}\right)+{cos}^{\pi^{{e}} } \left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$…
Question Number 77537 by john santu last updated on 07/Jan/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}\: \\ $$$$\mathrm{xy}'+\left(\mathrm{1}+\mathrm{xcot}\:\mathrm{x}\right)\mathrm{y}=\mathrm{x}\:? \\ $$ Answered by mr W last updated on 07/Jan/20 $${y}'+\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}\right){y}=\mathrm{1} \\…
Question Number 77473 by BK last updated on 06/Jan/20 Commented by mr W last updated on 06/Jan/20 $${y}=\frac{{dy}}{{dx}}+\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\frac{{d}^{\mathrm{3}} {y}}{{dy}^{\mathrm{3}} }+… \\ $$$$\frac{{dy}}{{dx}}=\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}}…
Question Number 11921 by carrot last updated on 05/Apr/17 $${given}\:{that}\:{y}={Acos}\mathrm{5}{x}\:+\:{Bsin}\mathrm{5}{x}, \\ $$$${show}\:{that}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\mathrm{25}{y}=\mathrm{0} \\ $$ Answered by sandy_suhendra last updated on 05/Apr/17 $$\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{5Asin5x}\:+\:\mathrm{5Bcos5x} \\…
Question Number 11887 by Peter last updated on 04/Apr/17 $$\frac{{dy}}{{dt}}\:+\mathrm{3}{t}^{\mathrm{2}} {y}\:=\:{t}^{\mathrm{2}\:} \:\:\:\:,\:{y}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$$${y}\left({t}\right)\:=\:? \\ $$ Answered by ajfour last updated on 04/Apr/17 $${when}\:\:\frac{{dy}}{{dt}}\:+{P}\:{y}\:={Q} \\…
Question Number 142935 by mnjuly1970 last updated on 07/Jun/21 Answered by qaz last updated on 07/Jun/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}}…
Question Number 11843 by tawa last updated on 02/Apr/17 $$\mathrm{Differentiate},\:\:\mathrm{ln}\left(\mathrm{cosx}\right)\:\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}. \\ $$ Answered by ajfour last updated on 02/Apr/17 $$\frac{{d}}{{dx}}\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\mathrm{cos}\:\left({x}+{h}\right)−\mathrm{ln}\:\mathrm{cos}\:{x}}{{h}} \\ $$$$\:=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left[\:\frac{\mathrm{cos}\:\left({x}+{h}\right)}{\mathrm{cos}\:{x}}\:\right]}{{h}} \\…
Question Number 142870 by mnjuly1970 last updated on 06/Jun/21 Answered by qaz last updated on 06/Jun/21 $$\Theta=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{u}\right)\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{u}}}{\mathrm{u}}\mathrm{du} \\ $$$$=\mathrm{2}\left\{\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{u}\right)\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{u}}}{\mathrm{u}}\mathrm{du}+\int_{\mathrm{0}}…
Question Number 142854 by liberty last updated on 06/Jun/21 $$\:\:{The}\:{maximum}\:{value}\:{of}\: \\ $$$$\:{y}=\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} }−\sqrt{{x}^{\mathrm{2}} +\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$$$\:{is}\: \\ $$ Commented by 1549442205PVT last…