Question Number 136295 by mnjuly1970 last updated on 20/Mar/21 $$\:\:\:\:\:\:….{nice}\:\:{calculus}… \\ $$$${prove}\::: \\ $$$$\mathrm{1}\:::\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{tan}^{−\mathrm{1}} \left(\sqrt{{tan}\left({x}\right)}\:\right)}{{tan}\left({x}\right)}{dx}=\frac{\pi}{\mathrm{2}}{log}\left(\mathrm{2}+\sqrt{\mathrm{2}}\:\right) \\ $$$$\mathrm{2}::\Omega=\int_{−\pi} ^{\:\pi} \frac{{e}^{\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)} {cos}\left({sin}\left({x}\right)\right)}{{e}^{{x}} +{e}^{{sin}\left({x}\right)} }{dx}=\pi \\…
Question Number 136284 by liberty last updated on 20/Mar/21 $${What}\:{is}\:{shortest}\:{distance}\: \\ $$$${from}\:{A}\left(\mathrm{3},\mathrm{8}\right)\:{to}\:{the}\:{circle}\: \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{6}{y}\:=\:\mathrm{12}\: \\ $$ Answered by EDWIN88 last updated on 20/Mar/21…
Question Number 70741 by aliesam last updated on 07/Oct/19 $${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\leqslant\mathrm{1}}\\{}\\{\mid{x}−\mathrm{2}\mid\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}>\mathrm{1}}\end{cases} \\ $$$$ \\ $$$${find}\:{the}\:{critical}\:{points} \\ $$ Commented by kaivan.ahmadi last updated on 07/Oct/19 $${f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}}…
Question Number 5171 by FilupSmith last updated on 25/Apr/16 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{show}\:\mathrm{why}: \\ $$$$\left({a}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{sin}\:{x}\right)=\mathrm{cos}\:{x} \\ $$$$\left({b}\right)\:\:\:\:\:\frac{{d}}{{dx}}\left(\mathrm{cos}\:{x}\right)=−\mathrm{sin}\:{x} \\ $$ Commented by FilupSmith last updated on 25/Apr/16 $${f}\:'\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}}…
Question Number 70705 by sadimuhmud 136 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{log}_{\mathrm{10}} \mathrm{x}\right)=? \\ $$ Answered by $@ty@m123 last updated on 07/Oct/19 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{log}_{{e}} \:{x}.}{\mathrm{log}_{{e}} \:\mathrm{10}}\right)=\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{10}}.\frac{{d}}{{dx}}\left(\mathrm{ln}\:{x}\right) \\…
Question Number 136225 by mnjuly1970 last updated on 19/Mar/21 $$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:\:{calculus}\:\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:{find}\:::\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {xe}^{−{x}} {coth}\left(\frac{{x}}{\mathrm{2}}\right){dx}=? \\ $$$$ \\ $$$$ \\ $$$$…
Question Number 70601 by oyemi kemewari last updated on 06/Oct/19 $$\mathrm{prove}\:\mathrm{thst}\:\mathrm{I}=\int_{\mathrm{0}} ^{\infty} \mathrm{f}\left(\mathrm{x}+\frac{\mathrm{1}}{×}\right)\frac{\mathrm{lnx}}{\mathrm{x}}\mathrm{dx}=\mathrm{0} \\ $$ Answered by mind is power last updated on 06/Oct/19 $${let}\:{x}=\frac{\mathrm{1}}{{y}}\Rightarrow{I}=\int_{\infty}…
Question Number 136138 by bramlexs22 last updated on 19/Mar/21 $${What}\:{is}\:{max}\:{and}\:{min}\:{value} \\ $$$${of}\:{y}=\mathrm{4}−{x}^{\mathrm{2}} −\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:? \\ $$ Answered by ajfour last updated on 19/Mar/21 $${let}\:\:\mathrm{1}−{x}^{\mathrm{2}} ={s}\geqslant\mathrm{0}…
Question Number 70582 by Raphael last updated on 05/Oct/19 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{y}=\frac{\mathrm{4}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)}}\:\mathrm{show}\:\mathrm{that}\:\mathrm{2}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{3x}^{\mathrm{2}} \mathrm{y}. \\ $$ Commented by kaivan.ahmadi last updated on 05/Oct/19 $$\mathrm{2}\left({x}^{\mathrm{3}} +\mathrm{1}\right)×\frac{−\frac{\mathrm{4}×\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}\sqrt{{x}^{\mathrm{3}}…
Question Number 70583 by Raphael last updated on 05/Oct/19 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{y}=\frac{\mathrm{4}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)}}\:\mathrm{show}\:\mathrm{that}\:\mathrm{2}\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=−\mathrm{3x}^{\mathrm{2}} \mathrm{y}. \\ $$ Answered by MJS last updated on 05/Oct/19 $${dy}=−\frac{\mathrm{6}{x}^{\mathrm{2}} }{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}}…