Question Number 18320 by Tinkutara last updated on 18/Jul/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC}\:\mathrm{with}\:\mathrm{fixed}\:\mathrm{base}\:{BC}, \\ $$$$\mathrm{the}\:\mathrm{vertex}\:{A}\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mathrm{cos}\:{B}\:+ \\ $$$$\mathrm{cos}\:{C}\:=\:\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:\frac{{A}}{\mathrm{2}}\:.\:\mathrm{If}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{the}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C} \\ $$$$\mathrm{respectively},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{b}\:+\:{c}\:=\:\mathrm{4}{a} \\ $$$$\left(\mathrm{2}\right)\:{b}\:+\:{c}\:=\:\mathrm{2}{a}…
Question Number 18318 by tawa tawa last updated on 18/Jul/17 $$\int\:\frac{\mathrm{x}\:+\:\mathrm{sinx}}{\mathrm{cosx}}\:\mathrm{dx} \\ $$ Commented by alex041103 last updated on 19/Jul/17 $${i}\:{think}\:{this}\:{is}\:{not}\:{elementary} \\ $$ Commented by…
Question Number 83852 by M±th+et£s last updated on 06/Mar/20 $${f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−\alpha{x}} {sin}\left({x}\right)}{{x}}{dx} \\ $$ Commented by mathmax by abdo last updated on 06/Mar/20 $${for}\:\alpha\geqslant\mathrm{0}\:\:{we}\:{have}\:{f}^{'}…
Question Number 149385 by liberty last updated on 05/Aug/21 Answered by Ar Brandon last updated on 05/Aug/21 $$\alpha+\frac{\mathrm{1}}{\alpha}=\mathrm{3}\Rightarrow\alpha^{\mathrm{2}} −\mathrm{3}\alpha+\mathrm{1}=\mathrm{0}\Rightarrow\alpha=\frac{\mathrm{3}\pm\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\Rightarrow\sqrt{\mathrm{tan}{x}}=\frac{\mathrm{3}\pm\sqrt{\mathrm{5}}}{\mathrm{2}}\Rightarrow\mathrm{tan}{x}=\frac{\mathrm{14}\pm\mathrm{6}\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{tan}^{\mathrm{3}} {x}+\frac{\mathrm{1}}{\mathrm{tan}^{\mathrm{3}} {x}}=\left(\frac{\mathrm{14}\pm\mathrm{6}\sqrt{\mathrm{5}}}{\mathrm{4}}\right)^{\mathrm{3}}…
Question Number 83850 by Rio Michael last updated on 06/Mar/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:{f},\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:{f}\left({x}\right)\:=\:\frac{{x}}{\mathrm{1}+\:{x}^{\mathrm{2}} }\:,\:{x}\in\mathbb{R} \\ $$ Commented by mathmax by abdo last updated on 06/Mar/20…
Question Number 83849 by Rio Michael last updated on 06/Mar/20 $$\mathrm{Gven}\:\mathrm{that}\:{y}\:=\:{e}^{−{x}} \mathrm{sin}{bx}\:,\mathrm{where}\:{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant},\mathrm{show}\:\mathrm{that} \\ $$$$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{2}\frac{{dy}}{{dx}}\:+\:\left(\mathrm{1}\:+\:{b}^{\mathrm{2}} \right){y}\:=\:\mathrm{0}. \\ $$ Commented by niroj last updated on…
Question Number 149383 by fotosy2k last updated on 05/Aug/21 Commented by fotosy2k last updated on 05/Aug/21 $${please}\:{help}\:{me} \\ $$ Answered by Ar Brandon last updated…
Question Number 149377 by fotosy2k last updated on 05/Aug/21 Answered by john_santu last updated on 05/Aug/21 $$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}.\mathrm{3}^{\mathrm{n}} }{\mathrm{3}^{\mathrm{n}} \left(\mathrm{1}+\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{n}} \right)}=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}}{\mathrm{1}+\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{n}} }=\mathrm{3} \\ $$…
Question Number 83842 by M±th+et£s last updated on 06/Mar/20 $$\int\frac{{ln}\left({x}\right)}{{ln}\left(\mathrm{6}{x}−{x}^{\mathrm{2}} \right)}{dx} \\ $$ Commented by Henri Boucatchou last updated on 06/Mar/20 $${Using}\:\:\:\frac{{lnA}}{{lnB}}=\frac{{A}}{{B}},\:\:\:\:\:\:\:\:\int\frac{{x}}{\mathrm{6}{x}−{x}^{\mathrm{2}} }{dx}=\int\left(−\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{2}{x}}{\mathrm{6}−{x}^{\mathrm{2}} }\right){dx}=−\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid\mathrm{6}−{x}^{\mathrm{2}} \mid\:+\:{Cte}…
Question Number 18307 by mondodotto@gmail.com last updated on 18/Jul/17 Answered by ajfour last updated on 18/Jul/17 $$\mathrm{Q}.\mathrm{2} \\ $$$$\mathrm{I}=\int_{\mathrm{0}} ^{\:\:\pi} \frac{\mathrm{ysin}\:\mathrm{ydy}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{y}}=\int_{\mathrm{0}} ^{\:\:\pi} \frac{\left(\pi−\mathrm{y}\right)\mathrm{sin}\:\mathrm{ydy}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{y}}…