Question Number 63372 by necx1 last updated on 03/Jul/19 $${For}\:{what}\:{values}\:{of}\:{a}\:{and}\:{b}\:{will}\:{the} \\ $$$${integral}\:\int_{{a}} ^{{b}} \sqrt{\mathrm{10}−{x}−{x}^{\mathrm{2}} }{dx}\:{be}\:{at} \\ $$$${maximum} \\ $$ Commented by mr W last updated…
Question Number 128907 by bramlexs22 last updated on 11/Jan/21 $$\:\mathrm{Given}\:\mathrm{U}_{\mathrm{n}} \:=\:\begin{cases}{\mathrm{2n}+\mathrm{1}\:;\:\mathrm{n}\:\mathrm{even}}\\{\mathrm{3n}+\mathrm{3}\:;\:\mathrm{n}\:\mathrm{odd}}\end{cases} \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{U}_{\mathrm{3}} +\mathrm{U}_{\mathrm{6}} +\mathrm{U}_{\mathrm{7}} +\mathrm{U}_{\mathrm{10}} +\mathrm{U}_{\mathrm{11}} + \\ $$$$\:\mathrm{U}_{\mathrm{14}} +…+\mathrm{U}_{\mathrm{27}} +\mathrm{U}_{\mathrm{28}} =? \\ $$…
Question Number 128903 by Engr_Jidda last updated on 11/Jan/21 $${solve}\:{the}\:{differential}\:{equation} \\ $$$$\frac{{dy}^{\mathrm{2}} }{{dx}^{\mathrm{2}\:} }+{y}=\mathrm{0} \\ $$ Answered by mindispower last updated on 11/Jan/21 $${X}^{\mathrm{2}} +\mathrm{1}=\mathrm{0}\Rightarrow{X}\in\left\{{i},−{i}\right\}…
Question Number 128900 by bemath last updated on 11/Jan/21 $$\:\int\:\frac{\mathrm{4x}+\mathrm{5}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)\left(\mathrm{x}+\mathrm{4}\right)\left(\mathrm{x}+\mathrm{5}\right)+\mathrm{1}}\:\mathrm{dx}?\: \\ $$ Answered by liberty last updated on 11/Jan/21 $$\:\int\:\frac{\mathrm{4x}+\mathrm{5}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{10}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{12}\right)+\mathrm{1}}\:\mathrm{dx}=\left(\ast\right) \\ $$$$\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7x}+\mathrm{10}\right)\left(\mathrm{x}^{\mathrm{2}}…
Question Number 63363 by rajesh4661kumar@gamil.com last updated on 03/Jul/19 Commented by Prithwish sen last updated on 03/Jul/19 $$\mathrm{tan70}=\frac{\mathrm{tan40}+\mathrm{tan20}+\mathrm{tan10}−\mathrm{tan40tan20tan10}}{\mathrm{1}−\mathrm{tan40tan20}−\mathrm{tan40tan10}−\mathrm{tan20tan10}} \\ $$$$\mathrm{tan70}\:−\:\mathrm{tan70tan40tan20}−\mathrm{tan70tan40tan10}−\mathrm{tan70tan20tan10} \\ $$$$=\:\mathrm{tan40}+\mathrm{tan20}+\mathrm{tan10}−\mathrm{tan40tan20tan10} \\ $$$$\because\mathrm{tan70tan20}=\mathrm{1}\:\therefore\mathrm{tan70tan40tan20}=\mathrm{tan40}\:\mathrm{and}\:\mathrm{tan70tan20tan10}\:=\:\mathrm{tan10} \\…
Question Number 128896 by ZiYangLee last updated on 11/Jan/21 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{even}\:\mathrm{obtaining}\:\mathrm{the}\:\mathrm{zero}\left(\mathrm{s}\right), \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{only}\:\mathrm{one}\:\mathrm{zero}. \\ $$$${f}\left({t}\right)=\left(\mathrm{1}+\sqrt{\mathrm{2}}{t}\right)\left(\mathrm{1}−{t}^{\mathrm{2}} \right)+{t}^{\mathrm{2}} \left({t}+\sqrt{\mathrm{2}}\right) \\ $$ Answered by Olaf last updated on 11/Jan/21…
Question Number 63361 by rajesh4661kumar@gamil.com last updated on 03/Jul/19 Commented by Prithwish sen last updated on 03/Jul/19 $$\mathrm{tan3A}\:=\:\mathrm{tan}\left(\mathrm{2A}+\mathrm{A}\right)=\frac{\mathrm{tan2A}+\mathrm{tanA}}{\mathrm{1}−\mathrm{tan2AtanA}} \\ $$$$\mathrm{tan3A}−\mathrm{tan3Atan2AtanA}\:=\:\mathrm{tan2A}+\mathrm{tanA} \\ $$$$\mathrm{tan3A}−\mathrm{tan2A}−\mathrm{tanA}\:=\:\mathrm{tan3Atan2AtanA}\:\mathrm{proved}. \\ $$ Commented…
Question Number 128892 by n0y0n last updated on 11/Jan/21 Commented by mr W last updated on 11/Jan/21 $${r}={distance}\:{from}\:{pedal}\:{point}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$$$\:\:\:\:\:\:\:{to}\:{a}\:{point}\:{C}\left({x},{y}\right)\:{on}\:{a}\:{curve} \\ $$$${p}={distance}\:{from}\:{pedal}\:{point}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$$$\:\:\:\:\:\:\:{to}\:{the}\:{tangent}\:{of}\:{the}\:{curve}\:{at}\:{the} \\…
Question Number 128893 by BHOOPENDRA last updated on 11/Jan/21 Answered by Dwaipayan Shikari last updated on 11/Jan/21 $$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {e}^{−\mathrm{3}{t}} {t}^{\mathrm{2}} {e}^{\mathrm{2}{t}} −{e}^{−\mathrm{3}{t}−\mathrm{2}{t}} {t}^{\mathrm{2}} {dt}\:\:\:\:\:\:\:\:\:\:{sinh}\mathrm{2}{t}=\frac{{e}^{\mathrm{2}{t}}…
Question Number 128888 by bemath last updated on 11/Jan/21 $$\:\int\:\frac{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}{\left(\mathrm{x}−\mathrm{4}\right)\left(\mathrm{x}−\mathrm{5}\right)\left(\mathrm{x}−\mathrm{6}\right)}\:\mathrm{dx}\:=? \\ $$ Answered by Olaf last updated on 11/Jan/21 $$\Omega\:=\:\int\frac{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)}{\left({x}−\mathrm{4}\right)\left({x}−\mathrm{5}\right)\left({x}−\mathrm{6}\right)}{dx} \\ $$$$\Omega\:=\:\int\left(\mathrm{1}+\frac{\mathrm{A}}{{x}−\mathrm{4}}+\frac{\mathrm{B}}{{x}−\mathrm{5}}+\frac{\mathrm{C}}{{x}−\mathrm{6}}\right){dx} \\ $$$$\mathrm{A}\:=\:\frac{\mathrm{3}×\mathrm{2}×\mathrm{1}}{\left(−\mathrm{1}\right)\left(−\mathrm{2}\right)}\:=\:\mathrm{3} \\…