Question Number 5115 by Yozzii last updated on 14/Apr/16 $${Given}\:{the}\:{space}\:{curve}\:\boldsymbol{{r}}=\boldsymbol{{r}}\left({t}\right),\:{show} \\ $$$${that}\:{its}\:{torsion}\:\tau\:{is}\:{given}\:{by} \\ $$$$\tau=\frac{\overset{.} {\boldsymbol{{r}}}\bullet\overset{..} {\boldsymbol{{r}}}×\overset{…} {\boldsymbol{{r}}}}{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid^{\mathrm{2}} }.\:{It}\:{may}\:{help}\:{to}\:{know}\:{that}\:{its} \\ $$$${curvature}\:{is}\:{numerically}\:{given}\:{by}\:\kappa=\frac{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid}{\mid\overset{.} {\boldsymbol{{r}}}\mid^{\mathrm{3}}…
Question Number 70651 by sadimuhmud 136 last updated on 06/Oct/19 Commented by Prithwish sen last updated on 06/Oct/19 $$\mathrm{Let}\:\sqrt{\mathrm{x}}\:=\:\mathrm{u}\:\:\Rightarrow\:\mathrm{d}\left(\sqrt{\mathrm{x}}\right)=\mathrm{du} \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{2}}{\:\sqrt{\mathrm{a}}}} \mathrm{e}^{\sqrt{\mathrm{a}}\mathrm{u}} \mathrm{du}\:\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{a}}}\:\left[\boldsymbol{\mathrm{e}}^{\sqrt{\boldsymbol{\mathrm{a}}}\boldsymbol{\mathrm{u}}} \right]_{\mathrm{0}}…
Question Number 5113 by FilupSmith last updated on 14/Apr/16 $$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}: \\ $$$$\left(\mathrm{1}\right)\:\:\mathrm{Focal}\:\mathrm{point} \\ $$$$\left(\mathrm{2}\right)\:\:\mathrm{Directrix} \\ $$$$\mathrm{for}\:{y}={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{simplicity},\:\mathrm{lets}\:\mathrm{assume}\:\mathrm{it}\:\mathrm{goes}\:\mathrm{throigh} \\ $$$$\mathrm{point}\:\left(\mathrm{0},\:\mathrm{0}\right). \\ $$…
Question Number 136186 by ZiYangLee last updated on 19/Mar/21 $$\mathrm{Let}\:{f}\left({x}\right)=\mathrm{cos}^{−\mathrm{1}} {x}+\mathrm{cos}^{−\mathrm{1}} \left({x}−\mathrm{1}\right),\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{f}. \\ $$ Answered by ZiYangLee last updated on 19/Mar/21 $$\mathrm{Ans}:\:\left[\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}}\right] \\…
Question Number 5111 by LMTV last updated on 13/Apr/16 $$\int\frac{{dy}}{{e}^{{y}} }=? \\ $$ Answered by FilupSmith last updated on 13/Apr/16 $$=\int{e}^{−{y}} {dy} \\ $$$$\int{e}^{{ax}+{b}} {dx}={ae}^{{ax}+{b}}…
Question Number 70638 by abdusalamyussif@gmail.com last updated on 06/Oct/19 Commented by MJS last updated on 06/Oct/19 $$\mathrm{the}\:\mathrm{first}\:\mathrm{question}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{complete} \\ $$ Answered by mr W last updated…
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Question Number 5100 by LMTV last updated on 12/Apr/16 $$\int\frac{{dy}}{\mathrm{1}+{y}^{\mathrm{2}} }=\mathrm{tan}^{−\mathrm{1}} {y}+{c} \\ $$$$\mathrm{why}??\:\mathrm{and} \\ $$$$\int\frac{{dy}}{\mathrm{1}−{y}^{\mathrm{2}} }=? \\ $$ Answered by Yozzii last updated on…
Question Number 5097 by LMTV last updated on 12/Apr/16 $${x}^{\mathrm{2}} {y}'={x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}} \\ $$$${yy}'+\mathrm{sin}\:{x}=\mathrm{0} \\ $$$${y}'=\mathrm{1}−{y}^{\mathrm{2}} \\ $$$$\mathrm{separation}\:\mathrm{of}\:\mathrm{variables} \\ $$ Answered by Yozzii last updated…
Question Number 136170 by mnjuly1970 last updated on 19/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:{calculus}…. \\ $$$$\:\:\:{compute}:: \\ $$$$\mathrm{2}{li}_{\mathrm{2}} \left(\frac{−\mathrm{1}}{\mathrm{2}}\right)−\mathrm{2}{li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)+{li}_{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)=?? \\ $$$$ \\ $$ Answered by mindispower last…