Question Number 83927 by M±th+et£s last updated on 08/Mar/20 $$\int\frac{{sinh}\left({x}\right)+{e}^{\mathrm{3}{x}} }{{sinh}\left({x}\right)−{e}^{{x}} }\:{dx} \\ $$ Answered by TANMAY PANACEA last updated on 08/Mar/20 $$\int\frac{\frac{{e}^{{x}} −{e}^{−{x}} }{\mathrm{2}}+{e}^{\mathrm{3}{x}}…
Question Number 149463 by jlewis last updated on 05/Aug/21 $$\mathrm{Let}\:\mathrm{the}\:\mathrm{independent}\:\mathrm{random}\:\mathrm{variables} \\ $$$$\mathrm{X}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{X}_{\mathrm{2}} \:\mathrm{have}\:\mathrm{binomial}\:\mathrm{distribution} \\ $$$$\mathrm{with}\:\mathrm{parameters}\:\mathrm{n}_{\mathrm{1}} =\mathrm{3},\mathrm{p}=\mathrm{2}/\mathrm{3}\:\mathrm{and}\:\mathrm{n}_{\mathrm{2}} =\mathrm{4} \\ $$$$\mathrm{p}=\mathrm{1}/\mathrm{2}\:\:\mathrm{respectively}.\: \\ $$$$\mathrm{Compute}\:\mathrm{P}\left(\mathrm{X}_{\mathrm{1}} =\mathrm{X}_{\mathrm{2}} \right) \\…
Question Number 149462 by liberty last updated on 05/Aug/21 Answered by Olaf_Thorendsen last updated on 05/Aug/21 $$\mathrm{A}\left(−\mathrm{3},\mathrm{0}\right)\:\mathrm{B}\left(\mathrm{1},−\mathrm{1}\right)\:\mathrm{C}\left(\mathrm{0},\mathrm{3}\right)\:\mathrm{D}\left(−\mathrm{1},\mathrm{3}\right) \\ $$$$\left(\mathrm{AC}\right)\::\:{y}\:=\:{x}+\mathrm{3} \\ $$$$\left(\mathrm{BD}\right)\::\:{y}\:=\:−\mathrm{2}{x}+\mathrm{1} \\ $$$$\mathrm{P}\:=\:\left(\mathrm{AC}\right)\cap\left(\mathrm{BD}\right)\:=\:\left(−\frac{\mathrm{2}}{\mathrm{3}},\frac{\mathrm{7}}{\mathrm{3}}\right) \\ $$$$\mathrm{P}\:=\:\left(\mid\mathrm{PA}\mid+\mid\mathrm{PB}\mid+\mid\mathrm{PC}\mid+\mid\mathrm{PD}\mid\right)_{{min}}…
Question Number 18388 by tawa tawa last updated on 19/Jul/17 $$\mathrm{Solve}\:\mathrm{simultaneously}.\: \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:=\:\mathrm{5}\:\:\:\:\:…….\:\left(\mathrm{i}\right) \\ $$$$\mathrm{5}^{\mathrm{x}} \:+\:\mathrm{y}\:=\:\mathrm{15}\:\:\:\:……\:\left(\mathrm{ii}\right) \\ $$ Answered by mrW1 last updated on 19/Jul/17…
Question Number 149457 by liberty last updated on 05/Aug/21 Answered by EDWIN88 last updated on 05/Aug/21 $$\:{f}\left({x}\right)=\mathrm{sin}\:^{\mathrm{4}} {x}−\mathrm{sin}\:{x}\mathrm{cos}\:{x}+\mathrm{cos}\:^{\mathrm{4}} {x} \\ $$$${f}\left({x}\right)=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x} \\ $$$${we}\:{define}\:\mathrm{sin}\:\mathrm{2}{x}={t} \\…
Question Number 83923 by jagoll last updated on 08/Mar/20 $$\int\:\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{20}}{\left(\mathrm{xsin}\:\mathrm{x}+\mathrm{5cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$ Answered by john santu last updated on 08/Mar/20 $$\Rightarrow\mathrm{xsin}\:\mathrm{x}+\mathrm{5cos}\:\mathrm{x}\:=\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{25}}\:\mathrm{cos}\:\left(\mathrm{x}−\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{5}}\right)\right)…
Question Number 18386 by Tinkutara last updated on 19/Jul/17 $$\mathrm{Prove}\:\mathrm{that}\:\sqrt[{\mathrm{3}}]{\mathrm{2}\:+\:\sqrt{\mathrm{5}}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{2}\:−\:\sqrt{\mathrm{5}}}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{rational}\:\mathrm{number}. \\ $$ Commented by mrW1 last updated on 19/Jul/17 $$\:\sqrt[{\mathrm{3}}]{\mathrm{2}\:+\:\sqrt{\mathrm{5}}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{2}\:−\:\sqrt{\mathrm{5}}}\:=\mathrm{1} \\ $$ Answered…
Question Number 18385 by Tinkutara last updated on 19/Jul/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{11}\:\mathrm{times}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$ Answered by mrW1 last updated on 19/Jul/17 $$\mathrm{let}'\mathrm{s}\:\mathrm{see}\:\mathrm{if}\:\mathrm{such}\:\mathrm{a}\:\mathrm{number}\:\mathrm{can}\:\mathrm{have}\:\mathrm{only} \\ $$$$\mathrm{one}\:\mathrm{digit}: \\…
Question Number 149458 by mathdanisur last updated on 05/Aug/21 Commented by EDWIN88 last updated on 06/Aug/21 $${n}\left({S}\right)={C}_{\mathrm{2}} ^{\mathrm{7}} =\frac{\mathrm{7}×\mathrm{6}}{\mathrm{2}×\mathrm{1}}=\mathrm{21} \\ $$$${n}\left({A}\right)=\:\mathrm{18} \\ $$$${A}=\left\{\left(\mathrm{1},\mathrm{2}\right),\left(\mathrm{1},\mathrm{4}\right),\left(\mathrm{1},\mathrm{6}\right),\left(\mathrm{2},\mathrm{3}\right),\left(\mathrm{2},\mathrm{4}\right),\right. \\ $$$$\:\:\:\:\:\left(\mathrm{2},\mathrm{5}\right),\left(\mathrm{2},\mathrm{6}\right),\left(\mathrm{3},\mathrm{4}\right),\left(\mathrm{3},\mathrm{6}\right),\left(\mathrm{4},\mathrm{5}\right),\left(\mathrm{4},\mathrm{6}\right),…
Question Number 83921 by jagoll last updated on 08/Mar/20 $$\int\:\:\frac{\mathrm{2x}^{\mathrm{12}} +\mathrm{5x}^{\mathrm{9}} }{\left(\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=\:? \\ $$ Answered by john santu last updated on 08/Mar/20…