Question Number 87013 by mathmax by abdo last updated on 01/Apr/20 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\left[{x}\right]} }{{x}+\mathrm{1}}{dx} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 152545 by liberty last updated on 29/Aug/21 Answered by Olaf_Thorendsen last updated on 31/Aug/21 $$\: \\ $$$$\mathrm{cos}\frac{\pi}{\mathrm{12}}\:=\:\mathrm{cos}\left(\frac{\pi}{\mathrm{4}}−\frac{\pi}{\mathrm{6}}\right) \\ $$$$\mathrm{cos}\frac{\pi}{\mathrm{12}}\:=\:\mathrm{cos}\frac{\pi}{\mathrm{4}}\mathrm{cos}\frac{\pi}{\mathrm{6}}+\mathrm{sin}\frac{\pi}{\mathrm{4}}\mathrm{sin}\frac{\pi}{\mathrm{6}} \\ $$$$\mathrm{cos}\frac{\pi}{\mathrm{12}}\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}.\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}.\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{cos}\frac{\pi}{\mathrm{12}}\:=\:\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}\sqrt{\mathrm{2}}}\:\:\:\:\left(\mathrm{1}\right)…
Question Number 152544 by liberty last updated on 29/Aug/21 $$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{real}\:\mathrm{zeros}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\:\mathrm{P}_{\mathrm{a}} \left(\mathrm{x}\right)=\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{ax}^{\mathrm{2}} \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{is}\:\mathrm{a}\:\mathrm{given}\:\mathrm{real}\:\mathrm{number} \\ $$ Commented by MJS_new last updated on…
Question Number 152546 by liberty last updated on 29/Aug/21 Answered by qaz last updated on 29/Aug/21 $$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}\centerdot\mathrm{3}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{7}\centerdot\mathrm{3}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{11}\centerdot\mathrm{3}^{\mathrm{5}} }+\frac{\mathrm{1}}{\mathrm{13}\centerdot\mathrm{3}^{\mathrm{6}} }−\frac{\mathrm{1}}{\mathrm{17}\centerdot\mathrm{3}^{\mathrm{8}} }−\frac{\mathrm{1}}{\mathrm{19}\centerdot\mathrm{3}^{\mathrm{9}} }+… \\ $$$$=\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{7}\centerdot\mathrm{3}^{\mathrm{3}}…
Question Number 87009 by mr W last updated on 01/Apr/20 $${solve} \\ $$$$\mathrm{7}\lfloor{x}+\mathrm{3}\rfloor^{\mathrm{2}} −\mathrm{3}\lfloor{x}\rfloor+\mathrm{6}=\mathrm{5}\:{mod}\:\mathrm{11} \\ $$ Answered by MJS last updated on 01/Apr/20 $$\lfloor{x}+\mathrm{3}\rfloor=\lfloor{x}\rfloor+\mathrm{3} \\…
Question Number 21471 by Joel577 last updated on 24/Sep/17 $$\mathrm{If}\:\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then} \\ $$$$\frac{\left({a}\:+\:{b}\right)\left({b}\:+\:{c}\right)\left({a}\:+\:{c}\right)}{{abc}}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:… \\ $$ Answered by Tinkutara last updated on 24/Sep/17 $${a}+{b}=−{c} \\ $$$$\therefore\frac{\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)}{{abc}}=\frac{\left(−{c}\right)\left(−{a}\right)\left(−{b}\right)}{{abc}}=−\mathrm{1} \\…
Question Number 21470 by Joel577 last updated on 24/Sep/17 $$\mathrm{2}^{{x}} \:=\:\mathrm{3}^{{y}} \:=\:\mathrm{6}^{−{z}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{2017}}{{x}}\:+\:\frac{\mathrm{2017}}{{y}}\:+\:\frac{\mathrm{2017}}{{z}}\right)^{\mathrm{2017}} \\ $$ Commented by Joel577 last updated on 24/Sep/17…
Question Number 152543 by liberty last updated on 29/Aug/21 $$\:\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\mathrm{x}^{\mathrm{3}} −\mathrm{3x}=\sqrt{\mathrm{x}+\mathrm{2}} \\ $$ Answered by EDWIN88 last updated on 29/Aug/21 $${it}\:{is}\:{clear}\:{for}\:{x}\geqslant−\mathrm{2} \\ $$$${case}\left(\mathrm{1}\right)\:−\mathrm{2}\leqslant{x}\leqslant\mathrm{2}\:,{let}\:{x}=\mathrm{2cos}\:{y}\:,\mathrm{0}\leqslant{y}\leqslant\pi…
Question Number 21469 by Tinkutara last updated on 24/Sep/17 $$\mathrm{Three}\:\mathrm{identical}\:\mathrm{blocks},\:\mathrm{each}\:\mathrm{having}\:\mathrm{a} \\ $$$$\mathrm{mass}\:{M},\:\mathrm{are}\:\mathrm{pushed}\:\mathrm{by}\:\mathrm{a}\:\mathrm{force}\:{F}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{frictionless}\:\mathrm{table}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{net}\:\mathrm{force} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{block}\:{A}? \\ $$ Commented by Tinkutara last updated on 24/Sep/17…
Question Number 152536 by Tawa11 last updated on 29/Aug/21 $$\mathrm{How}\:\mathrm{many}\:\mathrm{zeroes}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\:\:\mathrm{99}! \\ $$ Commented by MATHkingElsenK last updated on 29/Aug/21 $$\mathrm{99}/\mathrm{5}=\mathrm{19} \\ $$$$\mathrm{19}/\mathrm{5}=\mathrm{3} \\ $$$$\mathrm{19}+\mathrm{3}=\mathrm{22}\: \\…