Question Number 84957 by john santu last updated on 17/Mar/20 $$\int\:\left(\mathrm{2}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} \:\mathrm{dx}\:=\: \\ $$ Commented by john santu last updated on 17/Mar/20 $$\int\:\left(\mathrm{8}−\mathrm{3}\left(\mathrm{4x}^{\mathrm{2}} \right)+\mathrm{3}\left(\mathrm{2x}^{\mathrm{4}}…
Question Number 150489 by mathdanisur last updated on 12/Aug/21 $$\underset{\:\mathrm{0}} {\overset{\:\mathrm{2}} {\int}}\:\mid\mathrm{x}\mid\:\mathrm{x}^{\left[\boldsymbol{\mathrm{x}}+\mathrm{1}\right]} \:\mathrm{sgn}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 12/Aug/21 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}} \mid{x}\mid.{x}^{\left[{x}+\mathrm{1}\right]}…
Question Number 84954 by john santu last updated on 17/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}\:−\:\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\frac{\mathrm{5}}{\mathrm{2}}} }\:=\:? \\ $$ Answered by john santu last updated on 17/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}}…
Question Number 19419 by NEC last updated on 10/Aug/17 $$\mathrm{sin}\:\boldsymbol{{z}}=\mathrm{200} \\ $$$$ \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{z}} \\ $$ Commented by NEC last updated on 10/Aug/17 $${yes}.\:{i}\:{know}\:{that}. \\…
Question Number 19415 by Tinkutara last updated on 10/Aug/17 $${PS}\:\mathrm{is}\:\mathrm{a}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{of}\:\mathrm{length}\:\mathrm{4}\:\mathrm{and}\:{O} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{PS}.\:\mathrm{A}\:\mathrm{semicircular} \\ $$$$\mathrm{arc}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{with}\:{PS}\:\mathrm{as}\:\mathrm{diameter}.\:\mathrm{Let} \\ $$$${X}\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:\mathrm{this}\:\mathrm{arc}.\:{Q}\:\mathrm{and}\:{R} \\ $$$$\mathrm{are}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{arc}\:{PXS}\:\mathrm{such}\:\mathrm{that}\:{QR} \\ $$$$\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:{PS}\:\mathrm{and}\:\mathrm{the}\:\mathrm{semicircular} \\ $$$$\mathrm{arc}\:\mathrm{drawn}\:\mathrm{with}\:{QR}\:\mathrm{as}\:\mathrm{diameter}\:\mathrm{is} \\ $$$$\mathrm{tangent}\:\mathrm{to}\:{PS}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the} \\…
Question Number 19413 by Tinkutara last updated on 10/Aug/17 $$\mathrm{How}\:\mathrm{many}\:\mathrm{integer}\:\mathrm{pairs}\:\left({x},\:{y}\right)\:\mathrm{satisfy} \\ $$$${x}^{\mathrm{2}} \:+\:\mathrm{4}{y}^{\mathrm{2}} \:−\:\mathrm{2}{xy}\:−\:\mathrm{2}{x}\:−\:\mathrm{4}{y}\:−\:\mathrm{8}\:=\:\mathrm{0}? \\ $$ Commented by mrW1 last updated on 11/Aug/17 $$\left(\mathrm{0},−\mathrm{1}\right) \\…
Question Number 150486 by mathdanisur last updated on 12/Aug/21 $$\boldsymbol{\mathrm{x}}\:\in\:\mathbb{R} \\ $$$$\mid\mathrm{x}\:-\:\mathrm{1}\mid\:+\:\mid\mathrm{x}\:+\:\mathrm{3}\mid\:+\:\mid\mathrm{x}\:-\:\mathrm{5}\mid \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{given} \\ $$$$\mathrm{expression} \\ $$ Commented by MJS_new last updated on 12/Aug/21…
Question Number 19409 by Tinkutara last updated on 10/Aug/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:−\:\mathrm{7}\left[{x}\right]\:+\:\mathrm{5}\:=\:\mathrm{0}? \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\right. \\ $$$$\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x}.\:\mathrm{For}\:\mathrm{example} \\ $$$$\left.\left[\mathrm{3}.\mathrm{4}\right]\:=\:\mathrm{3}\:\mathrm{and}\:\left[−\mathrm{2}.\mathrm{3}\right]\:=\:−\mathrm{3}.\right) \\ $$ Commented by mrW1 last…
Question Number 150482 by mathdanisur last updated on 12/Aug/21 $$\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{4}} \:\frac{\boldsymbol{\pi}}{\mathrm{7}}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{4}} \:\frac{\mathrm{2}\boldsymbol{\pi}}{\mathrm{7}}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{4}} \:\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{7}}}\:\:=\:\:? \\ $$ Commented by liberty last updated on 13/Aug/21 $$=\mathrm{374} \\ $$…
Question Number 84942 by M±th+et£s last updated on 17/Mar/20 $$\int_{\mathrm{0}} ^{{x}} {sinh}\left({x}−{t}\right)\:{cosh}\left({t}\right)\:{dt} \\ $$ Answered by mind is power last updated on 17/Mar/20 $${sh}\left({x}\right){ch}\left({y}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left\{{sh}\left({x}+{y}\right)+{sh}\left({x}−{y}\right)\right\} \\…