Question Number 152273 by peter frank last updated on 27/Aug/21 $$\int\:\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int\frac{\mathrm{tan}\:\theta+\mathrm{tan}\:^{\mathrm{3}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{3}} \theta}\mathrm{d}\theta…
Question Number 152275 by peter frank last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:\mathrm{2xlog}\left(\:\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{sin}\:\mathrm{2x}\centerdot\mathrm{lntan}\:\mathrm{xdx}…
Question Number 86737 by M±th+et£s last updated on 30/Mar/20 $${prove}\:{that} \\ $$$$\mathrm{1}/{cos}\mathrm{2}{x}+{cosx}+\mathrm{1}=\frac{{sin}\frac{\mathrm{5}{x}}{\mathrm{2}}}{\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}/\frac{{cos}\left({x}\right)+{isin}\left({x}\right)−\mathrm{1}}{{cos}\left({x}\right)+{isin}\left({x}\right)+\mathrm{1}}=−{i}\:{tan}\left({x}\right) \\ $$$$ \\ $$$$\mathrm{3}/\frac{{cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right)+\mathrm{1}}{{cos}\left(\mathrm{5}{x}\right)−{isin}\left({x}\right)+\mathrm{1}}={cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right) \\ $$ Commented by som(math1967) last updated…
Question Number 21200 by Tinkutara last updated on 15/Sep/17 $$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$ Commented by mrW1 last updated…
Question Number 86734 by Tony Lin last updated on 30/Mar/20 $${Find}\:{all}\:{functions}\:{that}\:{satisfy}\:{the} \\ $$$${equation} \\ $$$$\left[\int{f}\left({x}\right){dx}\right]\left[\int\frac{\mathrm{1}}{{f}\left({x}\right)}{dx}\right]=−\mathrm{1} \\ $$ Answered by mr W last updated on 30/Mar/20…
Question Number 152271 by peter frank last updated on 27/Aug/21 $$\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\:\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\mathrm{A}=\int\left(\mathrm{3x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int\left(\mathrm{2x}+\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}}…
Question Number 152270 by peter frank last updated on 27/Aug/21 $$\int\frac{\mathrm{5x}+\mathrm{3}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{10}}}\mathrm{dx} \\ $$ Answered by Olaf_Thorendsen last updated on 27/Aug/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{\mathrm{5}{x}+\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}\:{dx} \\ $$$$\mathrm{F}\left({u}−\mathrm{2}\right)\:=\:\int\frac{\mathrm{5}{u}−\mathrm{7}}{\:\sqrt{{u}^{\mathrm{2}}…
Question Number 152265 by mathdanisur last updated on 26/Aug/21 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{k}}\:-\:\sqrt{\mathrm{k}\:-\:\mathrm{1}}}{\:\sqrt{\mathrm{k}\left(\mathrm{k}\:+\:\mathrm{1}\right.}}\right)\:=\:? \\ $$ Answered by mindispower last updated on 27/Aug/21 $${sin}^{−} \left({a}\right)−{sin}^{−} \left({b}\right)={sin}^{−}…
Question Number 21194 by vivek last updated on 15/Sep/17 $$.\:\underset{{x}\rightarrow\mathrm{2}^{+} } {\mathrm{li}{m}}\:\left\{\frac{\left[{x}\right]^{\mathrm{3}} }{\mathrm{3}}\:−\left[\:\frac{{x}}{\mathrm{3}}\right]^{\mathrm{3}} \right\}\:{is}\:{equal}\:{to}\:… \\ $$$$ \\ $$ Commented by vivek last updated on 16/Sep/17…
Question Number 86728 by M±th+et£s last updated on 30/Mar/20 $$\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+\frac{{b}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)\:{dx} \\ $$ Commented by mathmax by abdo last updated on 30/Mar/20…