Question Number 115260 by bobhans last updated on 24/Sep/20 $${Let}\:{f}\:{be}\:{a}\:{real}\:{valued}\:{function}\:{defined} \\ $$$${on}\:{the}\:{interval}\:\left(−\mathrm{1},\mathrm{1}\right)\:{such}\:{that}\: \\ $$$${e}^{−{x}} .{f}\left({x}\right)=\mathrm{2}+\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\sqrt{{t}^{\mathrm{4}} +\mathrm{1}}\:{dt}\:\forall{x}\in\left(−\mathrm{1},\mathrm{1}\right) \\ $$$${and}\:{let}\:{g}\:{be}\:{the}\:{inverse}\:{function}\:{of}\:{f} \\ $$$$.\:{Find}\:{the}\:{value}\:{of}\:{g}'\left(\mathrm{2}\right). \\ $$ Commented…
Question Number 115222 by mnjuly1970 last updated on 24/Sep/20 $$\:\:\:\:\:\:….\:{nice}\:\:{math}\:… \\ $$$$ \\ $$$$\:\:\:\:{nice}\:\:{integral}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\::: \\ $$$$\Psi=\mathrm{9}\int_{\mathrm{0}} ^{\:\infty} {x}^{\mathrm{5}} {e}^{−{x}^{\mathrm{3}} } {ln}\left(\mathrm{1}+{x}\right){dx}\:\overset{???} {=}\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)−\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)+\Gamma\left(\frac{\mathrm{3}}{\mathrm{3}}\right)\:\: \\…
Question Number 180718 by a.lgnaoui last updated on 16/Nov/22 $${Resoudre}\:{le}\:{systeme} \\ $$$$\frac{{dx}}{{dt}}=\mathrm{4}{x}+\mathrm{6}{y} \\ $$$$\frac{{dy}}{{dt}}=−\mathrm{3}{x}−\mathrm{5}{y} \\ $$$$\frac{{dz}}{{dt}}=−\mathrm{3}{x}−\mathrm{6}{y}−\mathrm{5}{z} \\ $$ Answered by mr W last updated on…
Question Number 115112 by bemath last updated on 23/Sep/20 $${What}\:{is}\:{minimum}\:{distance}\:{between}\: \\ $$$${xy}\:=\:\mathrm{4}\:{and}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:=\:\mathrm{4}\:?\: \\ $$$$ \\ $$ Answered by bobhans last updated on 28/Sep/20…
Question Number 114996 by mnjuly1970 last updated on 22/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{mathematics}…\: \\ $$$$\:\:\:\:{prove}\:{that}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{sinh}^{\mathrm{2}} \left(\pi{n}\right)}\:=\frac{\mathrm{1}}{\mathrm{6}}\:−\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{8}\pi}\:\:\checkmark\checkmark \\…
Question Number 114945 by bemath last updated on 22/Sep/20 Answered by bobhans last updated on 22/Sep/20 $${f}\left({x},{y}\right)\:=\:{x}^{\mathrm{3}} +{y}^{\mathrm{3}} −\mathrm{3}{x}−\mathrm{12}{y}+\mathrm{20} \\ $$$${f}_{{x}} =\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0}\:\rightarrow\begin{cases}{{x}=\mathrm{1}}\\{{x}=−\mathrm{1}}\end{cases} \\ $$$${f}_{{y}}…
Question Number 114922 by bemath last updated on 22/Sep/20 $${find}\:{minimum}\:{value}\:{of}\:{function} \\ $$$${y}=\sqrt{\left({x}+\mathrm{6}\right)^{\mathrm{2}} +\mathrm{25}}\:+\sqrt{\left({x}−\mathrm{6}\right)^{\mathrm{2}} +\mathrm{121}} \\ $$ Answered by john santu last updated on 22/Sep/20 $${you}\:{want}\:{to}\:{find}\:{the}\:{point}\:{on}\:{the}\:…
Question Number 180350 by mnjuly1970 last updated on 10/Nov/22 Answered by Ar Brandon last updated on 11/Nov/22 $$\mathcal{L}\left({e}^{−{x}} .{erf}\left(\sqrt{{x}}\right)\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {erf}\left(\sqrt{{x}}\right){e}^{−{px}} {dx} \\ $$$$=\int_{\mathrm{0}}…
Question Number 114753 by bemath last updated on 21/Sep/20 $${find}\:{minimum}\:{value}\:{of}\:{function} \\ $$$${f}\left({x}\right)\:=\:\frac{\left({x}+\mathrm{17}\right)^{\mathrm{3}} }{{x}}\:,\:{x}>\mathrm{0} \\ $$ Answered by Olaf last updated on 21/Sep/20 $${f}'\left({x}\right)\:=\:\frac{\mathrm{3}\left({x}+\mathrm{17}\right)^{\mathrm{2}} {x}−\left({x}+\mathrm{17}\right)^{\mathrm{3}} }{{x}^{\mathrm{2}}…
Question Number 114735 by mnjuly1970 last updated on 20/Sep/20 $$\:\:\:….\:{nice}\:\:{mathematics}… \\ $$$$ \\ $$$$\:\:{prove}\:\:{that}::\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left[\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}\right]^{\mathrm{2}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)\mathrm{2}^{\mathrm{4}{n}} }\:=\mathrm{1}−\frac{\mathrm{2}}{\pi}\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{m}.{n}.{july}.\:\mathrm{1970}# \\…