Question Number 62874 by solihin last updated on 26/Jun/19 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\left(\mathrm{6}{x}\right)}{{tan}\left(\mathrm{5}{x}\right)} \\ $$$$ \\ $$$${how}\:\:{to}\:\:{solve}\:\:{this}\:\:{w}/{o}\:\:{L}'{hospital}'{s}\:\:{rule}? \\ $$ Commented by kaivan.ahmadi last updated on 26/Jun/19 $${lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 128408 by bramlexs22 last updated on 07/Jan/21 $$\rho\:=\:\int\:\frac{\mathrm{sin}\:\left(\mathrm{4}{x}\right)}{\mathrm{sin}\:^{\mathrm{4}} \left({x}\right)+\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)}\:{dx}\: \\ $$ Answered by mr W last updated on 07/Jan/21 $$=\int\frac{\mathrm{sin}\:\left(\mathrm{4}{x}\right)}{\mathrm{1}−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}^{\mathrm{2}} \:{x}}{dx}…
Question Number 62869 by aliesam last updated on 26/Jun/19 $${y}\frac{{dy}}{{dx}}\:−\:\frac{{y}}{\frac{{dy}}{{dx}}}\:=\:\mathrm{2}{a} \\ $$$$ \\ $$$${a}\:{is}\:{a}\:{real}\:{number} \\ $$ Answered by Hope last updated on 26/Jun/19 $${y}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} −{y}=\mathrm{2}{a}\left(\frac{{dy}}{{dx}}\right)…
Question Number 128402 by bramlexs22 last updated on 07/Jan/21 $$\:\mathrm{If}\:\mathrm{f}\left({x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{4}\right)={x}+\mathrm{3}\:\mathrm{then}\:\mathrm{f}\left(−\mathrm{2}\right)=? \\ $$ Answered by liberty last updated on 07/Jan/21 $$\Rightarrow\mathrm{x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{4}=−\mathrm{2}\:;\:\mathrm{x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{6}=\mathrm{0} \\ $$$$\:\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}−\mathrm{2}\right)=\mathrm{0}\:\rightarrow\begin{cases}{\mathrm{x}=\mathrm{3}}\\{\mathrm{x}=\mathrm{2}}\end{cases}\:\Leftrightarrow\mathrm{f}\left(−\mathrm{2}\right)=\begin{cases}{\mathrm{6}}\\{\mathrm{5}}\end{cases}…
Question Number 128401 by bramlexs22 last updated on 07/Jan/21 $$\mathrm{Given}\:\mathrm{function}\:\mathrm{f}\left({x}+\mathrm{1}\right)+\mathrm{f}\left({x}−\mathrm{1}\right)={x}^{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{f}^{−\mathrm{1}} \left({x}\right)\:=\:? \\ $$$$\left({A}\right)\:\pm\sqrt{\mathrm{1}−\mathrm{2}{x}}\:;\:{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({B}\right)\:\pm\sqrt{{x}+\mathrm{2}}\:;\:{x}\geqslant−\mathrm{2} \\ $$$$\left({C}\right)\:\pm\sqrt{\mathrm{2}{x}+\mathrm{1}}\:;\:{x}\geqslant−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({D}\right)\:\pm\sqrt{\mathrm{3}{x}−\mathrm{1}}\:;\:{x}\geqslant\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\left({E}\right)\:\pm\sqrt{\mathrm{2}{x}−\mathrm{1}}\:;\:{x}\geqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$…
Question Number 62861 by ajfour last updated on 26/Jun/19 Commented by mr W last updated on 26/Jun/19 $${very}\:{hard}\:{task}\:{sir},\:{maybe}\:{not} \\ $$$${possible}\:{to}\:{solve}. \\ $$ Commented by ajfour…
Question Number 128395 by bramlexs22 last updated on 06/Jan/21 $$\:\:\:\frac{{dy}}{{dx}}\:=\:\left(\frac{{x}}{{y}}\right)\:\mathrm{ln}\:\left(\frac{{x}}{{y}}\right) \\ $$ Answered by mr W last updated on 07/Jan/21 $${let}\:{x}={yu} \\ $$$$\frac{{dx}}{{dy}}={u}+{y}\frac{{du}}{{dy}} \\ $$$$\mathrm{1}=\left({u}+{y}\frac{{du}}{{dy}}\right){u}\mathrm{ln}\:{u}…
Question Number 62856 by mathmax by abdo last updated on 26/Jun/19 $${let}\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}} }\:{dx}\:\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{also}\:{g}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{4}} }{\left({x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}}…
Question Number 62855 by mathmax by abdo last updated on 26/Jun/19 $${find}\:\int\:\:\left(\frac{{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\right)^{\mathrm{2}} \:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{8}} }{\left(\mathrm{1}+{x}^{\mathrm{6}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty}…
Question Number 128388 by enter last updated on 06/Jan/21 $${y}=−{sin}\left(\frac{\Pi}{\mathrm{2}}+\mathrm{2}{x}\right)+\mathrm{2}{cos}\left(\mathrm{5}{x}+\Pi\right) \\ $$$${y}_{{min}} =\:?? \\ $$ Commented by TheSupreme last updated on 07/Jan/21 $${y}=−{cos}\left(\mathrm{2}{x}\right)−\mathrm{2}{cos}\left(\mathrm{5}{x}\right)\geqslant−\mathrm{3} \\ $$$${y}=−\mathrm{3}\:{x}=\mathrm{10}{k}\pi…