Question Number 63552 by minh2001 last updated on 05/Jul/19 $${Calculate}\:\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\mathrm{2}}} {\int}}{x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}+\underset{\frac{\mathrm{1}}{\mathrm{2}}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}+\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\:{dx}+\underset{\mathrm{2}} {\overset{\mathrm{3}} {\int}}{x}^{\mathrm{4}} \sqrt{{x}^{\mathrm{5}} +\mathrm{1}\:}{dx}+…+\underset{\mathrm{78}}…
Question Number 129086 by Ar Brandon last updated on 12/Jan/21 $$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\left(\frac{\mathrm{e}^{\mathrm{x}} −\mathrm{1}}{\mathrm{x}}\right) \\ $$ Answered by Dwaipayan Shikari last updated on 12/Jan/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{log}\left(\frac{{e}^{{x}}…
Question Number 129087 by Adel last updated on 12/Jan/21 Answered by MJS_new last updated on 12/Jan/21 $$\mathrm{2}^{\mathrm{10}} \mathrm{10}!\left(\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×\mathrm{11}×\mathrm{13}×\mathrm{15}×\mathrm{17}×\mathrm{19}\right)= \\ $$$$=\mathrm{10}!×\mathrm{11}×\left(\mathrm{3}×\mathrm{2}^{\mathrm{2}} \right)×\mathrm{13}×\left(\mathrm{7}×\mathrm{2}\right)×\mathrm{15}×\left(\mathrm{2}^{\mathrm{4}} \right)×\mathrm{17}×\left(\mathrm{9}×\mathrm{2}\right)×\mathrm{19}×\left(\mathrm{5}×\mathrm{2}^{\mathrm{2}} \right)= \\ $$$$=\mathrm{20}!…
Question Number 129084 by mnjuly1970 last updated on 12/Jan/21 $$\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{20}}+\frac{\mathrm{1}}{\mathrm{56}}−\frac{\mathrm{1}}{\mathrm{120}}+\frac{\mathrm{1}}{\mathrm{220}}−…\right) \\ $$$$\:\:\:\:\:\:\:\overset{???} {=}\pi\:−\mathrm{3}\:\: \\ $$ Answered by Dwaipayan Shikari last updated…
Question Number 129077 by Study last updated on 12/Jan/21 Commented by benjo_mathlover last updated on 12/Jan/21 $$\frac{\mathrm{FG}}{\mathrm{8}}\:=\:\frac{\mathrm{5}}{\mathrm{6}}\Rightarrow\mathrm{FG}=\frac{\mathrm{20}}{\mathrm{3}} \\ $$$$\frac{\mathrm{DE}}{\mathrm{8}}=\frac{\mathrm{3}}{\mathrm{6}}\Rightarrow\mathrm{DE}=\mathrm{4} \\ $$$$\mathrm{Area}\:\mathrm{FGED}\:=\:\frac{\mathrm{8}×\mathrm{6}}{\mathrm{2}}−\frac{\frac{\mathrm{20}}{\mathrm{3}}+\mathrm{8}}{\mathrm{2}}×\mathrm{1}−\frac{\mathrm{3}×\mathrm{4}}{\mathrm{2}}\: \\ $$$$\:=\:\mathrm{24}−\left(\frac{\mathrm{10}}{\mathrm{3}}+\mathrm{4}\right)−\mathrm{6} \\ $$$$\:=\:\mathrm{18}−\frac{\mathrm{22}}{\mathrm{3}}\:=\:\frac{\mathrm{32}}{\mathrm{3}}…
Question Number 129072 by liberty last updated on 12/Jan/21 $$\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{F}\left(\mathrm{0}\right)\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\: \\ $$$$\:\mathrm{F}\left(\mathrm{x}\right)=\:\frac{\left(\mathrm{4}^{{x}} −\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{4}}\right)\:\mathrm{ln}\:\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\right)}\:\mathrm{becomes}\:\mathrm{continous} \\ $$$$\mathrm{at}\:{x}\:=\:\mathrm{0}? \\ $$ Answered by MJS_new last updated on…
Question Number 129073 by amns last updated on 12/Jan/21 $$\boldsymbol{{Prove}}\:\boldsymbol{{that}}: \\ $$$${sec}^{\mathrm{4}} \theta\:−\:{sec}^{\mathrm{2}} \theta\:=\:{tan}^{\mathrm{4}} \theta\:+\:{tan}^{\mathrm{2}} \theta \\ $$$${Please}\:{help}\:{me}\:{quickly}! \\ $$ Commented by benjo_mathlover last updated…
Question Number 63536 by Rio Michael last updated on 05/Jul/19 $$\left.{a}\right)\:\:{if}\:{y}=\:{x}^{{m}} \left(\mathrm{1}−{x}\right)^{{n}} ,\:{where}\:{n}\in\:\mathbb{Z}^{+} ,\:{the}\:{set}\:{of}\:{positive}\:{integers}, \\ $$$${show}\:{that}\:{when}\:\frac{{dy}}{{dx}}=\mathrm{0},\:{x}=\frac{{m}}{{m}+{n}} \\ $$$$\left.{b}\right){if}\:{y}\:=\:\mathrm{2}\left({x}−\mathrm{5}\right)\sqrt{{x}+\mathrm{4}}\:,{show}\:{that}\:\frac{{dy}}{{dx}}\:=\:\frac{\mathrm{3}\left({x}+\mathrm{1}\right)}{\:\sqrt{{x}+\mathrm{4}}\:} \\ $$$$\left.{c}\right)\:{solve}\:{the}\:{equation}\:\:{sinx}−{sin}\mathrm{5}{x}+{cos}\mathrm{3}{x}\:=\:\mathrm{0}\:{for}\:\:\mathrm{0}°\leqslant{x}\leqslant\mathrm{180}° \\ $$ Commented by Prithwish…
Question Number 129070 by benjo_mathlover last updated on 12/Jan/21 $$\:\mathrm{Solve}\:\mathrm{2cos}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{2x}\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{3}}\:−\mathrm{3x}\right) \\ $$ Answered by liberty last updated on 12/Jan/21 $$\:\Leftrightarrow\:\mathrm{2cos}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{2x}\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{3}}−\mathrm{3x}\right) \\ $$$$\:\Leftrightarrow\:\mathrm{cos}\:\mathrm{3x}+\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{3x}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{sin}\:\mathrm{3x} \\ $$$$\:\Leftrightarrow\:\mathrm{cos}\:\mathrm{x}\:=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{sin}\:\mathrm{3x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{3x}\: \\…
Question Number 63534 by Rio Michael last updated on 05/Jul/19 $${find}\:{the}\:{set}\:{of}\:{values}\:{of}\:{x}\:{for}\:{which}\:{y}\:{is}\:{real}\:{if}\: \\ $$$$\:{y}=\frac{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{1}\right)}{{x}+\mathrm{2}}\:,\:{x}\neq−\mathrm{2},\:{x}\in\mathbb{R} \\ $$ Answered by MJS last updated on 05/Jul/19 $$\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{already}\:\mathrm{given}:\:{x}\in\mathbb{R}\backslash\left\{−\mathrm{2}\right\} \\ $$…