Question Number 165581 by mnjuly1970 last updated on 04/Feb/22 $$ \\ $$$$\varphi\left({t}\right)=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:{sin}\left({x}\right)+{t}\:{cos}\left({x}\right)\right)^{\:\mathrm{2}} {dx} \\ $$$${find}\:\:{the}\:\:{value}\:{of}\:{the}\:{extermum} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{of}\:\:\:\varphi\:\left({t}\right). \\ $$ Answered by aleks041103 last…
Question Number 100032 by bobhans last updated on 24/Jun/20 $$\mathrm{Given}\:\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}+\mathrm{1}}\right)\:=\:\mathrm{x}^{\mathrm{2}} \:.\:\mathrm{find}\:\mathrm{minimum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{function}\:\mathrm{h}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)−\frac{\mathrm{3}}{\mathrm{x}−\mathrm{1}} \\ $$ Commented by john santu last updated on 24/Jun/20 $$\mathrm{let}\:\frac{\mathrm{x}}{\mathrm{x}+\mathrm{1}}\:=\:\mathrm{z}\:\Rightarrow\mathrm{xz}+\mathrm{z}\:=\:\mathrm{x} \\…
Question Number 99960 by bemath last updated on 24/Jun/20 $$\mathrm{Given}\:\mathrm{y}\sqrt{\mathrm{x}}+\mathrm{x}\sqrt{\mathrm{y}}\:=\:\mathrm{2}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}\:\mid_{\left(\mathrm{1},\mathrm{1}\right)} \:=\:?\: \\ $$ Commented by Dwaipayan Shikari last updated on 24/Jun/20 $$−\mathrm{1} \\…
Question Number 165471 by leonhard77 last updated on 02/Feb/22 $$\:\begin{cases}{{h}\left(\mathrm{3}{x}\right)=\left(\frac{\mathrm{2}−{x}}{{x}+\mathrm{1}}−{f}\left({x}^{\mathrm{3}} \right)\right)^{\mathrm{2}} }\\{{f}\left(\mathrm{1}\right)={f}\:'\left(\mathrm{1}\right)=\mathrm{2}}\end{cases} \\ $$$$\:{h}\:'\left(\mathrm{3}\right)=? \\ $$ Commented by cortano1 last updated on 04/Feb/22 $$\:\mathrm{3}{h}'\left(\mathrm{3}{x}\right)=\mathrm{2}\left(\frac{−{x}+\mathrm{2}}{{x}+\mathrm{1}}−{f}\left({x}^{\mathrm{3}} \right)\right)\left(\frac{−\mathrm{3}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}}…
Question Number 165441 by mnjuly1970 last updated on 01/Feb/22 Answered by mahdipoor last updated on 01/Feb/22 $$\mathrm{if}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{increment} \\ $$$$\mathrm{function}\:\mathrm{in}\:\mathrm{x}\geqslant\mathrm{a}\:\Rightarrow\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}}\lfloor\mathrm{g}\left(\mathrm{x}\right)\rfloor=\lfloor\mathrm{g}\left(\mathrm{a}\right)\rfloor \\ $$$${prove}:\:\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}g}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{a}\right)\Rightarrow\forall\varepsilon>\mathrm{0}\:\:\:\exists\sigma>\mathrm{0}\:\:,…
Question Number 165328 by mnjuly1970 last updated on 30/Jan/22 $$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\mathscr{N}{ice}\:\:\:\mathscr{I}{ntegral} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}^{\:\frac{\mathrm{3}}{\mathrm{2}}} \right)}{{x}^{\:\mathrm{2}} }\:{dx}\:\:=\frac{\pi\:+\:\sqrt{\mathrm{3}}\:{ln}\left(\mathrm{7}\:+\mathrm{4}\sqrt{\mathrm{3}}\:\right)}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:−−−−−−−−−\:\:\:…
Question Number 165194 by mnjuly1970 last updated on 27/Jan/22 $$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} {ln}\:\left(\:\mathrm{1}+\:{cos}\:\left({x}\right)\right).{cos}\:\left({nx}\:\right){dx}=? \\ $$ Answered by mindispower last updated on 27/Jan/22 $$=\int_{\mathrm{0}} ^{\mathrm{2}\pi}…
Question Number 99646 by 24224 Opiyo Kamuki last updated on 22/Jun/20 $$\boldsymbol{{use}}\:\boldsymbol{{power}}\:\boldsymbol{{series}}\:\boldsymbol{{solution}}\:\boldsymbol{{method}}\:\boldsymbol{{to}}\:\boldsymbol{{solve}}\:\boldsymbol{{the}}\:\boldsymbol{{ODE}} \\ $$$$\boldsymbol{{y}}''−\boldsymbol{{xy}}=\mathrm{0} \\ $$ Answered by MWSuSon last updated on 22/Jun/20 $$\underset{\mathrm{k}=\mathrm{2}} {\overset{\infty}…
Question Number 165168 by saboorhalimi last updated on 26/Jan/22 Answered by mahdipoor last updated on 26/Jan/22 $$\frac{{dy}}{{dx}}=\frac{{dy}/{dt}}{{dx}/{dt}}=\frac{\mathrm{3}{t}^{\mathrm{2}} }{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{4}}=\mathrm{1}+\frac{\mathrm{4}}{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{4}} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{{d}\left({dy}/{dx}\right)}{{dx}}=\frac{\frac{{d}\left({dy}/{dx}\right)}{{dt}}}{\frac{{dx}}{{dt}}}=\frac{\frac{−\mathrm{4}\left(\mathrm{6}{t}\right)}{\left(\mathrm{3}{t}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{2}}…
Question Number 165152 by mnjuly1970 last updated on 26/Jan/22 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{prove} \\ $$$$\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}\:−\:{x}^{\:\mathrm{2}} }{\left(\mathrm{1}+{x}\:\right){ln}\left({x}\right)}\:{dx}\:=\:{ln}\left(\frac{\mathrm{4}}{\pi}\:\right) \\ $$$$\:\:\:−−−−− \\ $$ Answered by mindispower last…