Question Number 150539 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{the}\:\mathrm{foolowing}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}+\mathrm{2}}{\mathrm{2}}} \:\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:=\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{e}^{−\boldsymbol{\pi\mathrm{n}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \\ $$ Answered by…
Question Number 150533 by mnjuly1970 last updated on 13/Aug/21 $$\:\:\:\: \\ $$$$\:\:\:\:\mathrm{solve}… \\ $$$$\:\:\mathrm{I}:=\:\int_{−\infty} ^{\:\infty} {e}^{\:{x}−{n}.{sinh}^{\:\mathrm{2}} \left({x}\right)} {dx}\:\overset{??} {=}\sqrt{\frac{\pi}{{n}}} \\ $$$$\:\:\:\:\:\:\:…{m}.{n}… \\ $$ Answered by…
Question Number 84998 by jagoll last updated on 18/Mar/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{29}} \\ $$$$\mathrm{in}\:\mathrm{expression}\:\left(\mathrm{1}+\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{7}} +\mathrm{x}^{\mathrm{9}} \right)^{\mathrm{29}} \\ $$ Answered by mind is power last updated on…
Question Number 84997 by M±th+et£s last updated on 18/Mar/20 $${log}_{\frac{{x}}{\mathrm{2}}} {x}^{\mathrm{2}} −{log}_{\mathrm{16}{x}} {x}^{\mathrm{3}} +\mathrm{40}{log}_{\mathrm{4}{x}} \sqrt{{x}}=\mathrm{0} \\ $$ Commented by jagoll last updated on 18/Mar/20 $$\frac{\mathrm{log}_{\mathrm{2}}…
Question Number 150531 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Find}\:\:\mathrm{x};\mathrm{y}\:\:;\:\:\mathrm{x}\in\mathrm{Q}\:\:\mathrm{and}\:\:\mathrm{y}\in\mathrm{Z}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\mathrm{2020}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)\:+\:\mathrm{2019}\left(\mathrm{x}\:+\:\mathrm{y}\right)\:=\:\mathrm{2021xy} \\ $$ Commented by Rasheed.Sindhi last updated on 14/Aug/21 $$\underset{\smile} {\overset{\frown}…
Question Number 19457 by NEC last updated on 11/Aug/17 $${prove}\:{that}\:\int\mathrm{tan}\:^{\mathrm{2}} {xdx} \\ $$$$ \\ $$$$=\mathrm{tan}\:{x}−{x} \\ $$ Answered by Joel577 last updated on 11/Aug/17 $$\int\:\mathrm{tan}^{\mathrm{2}}…
Question Number 84993 by mr W last updated on 18/Mar/20 $$\mathrm{100}\:{apples}\:{should}\:{be}\:{packed}\:{in}\:{three} \\ $$$${boxes}\:{and}\:{each}\:{box}\:{should}\:{contain} \\ $$$${at}\:{least}\:\mathrm{10}\:{apples}.\:{in}\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}? \\ $$ Commented by john santu last updated…
Question Number 19455 by Tinkutara last updated on 11/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{z}\:=\:\mathrm{cos}\:\mathrm{6}°\:+\:{i}\:\mathrm{sin}\:\mathrm{6}°,\:\mathrm{then} \\ $$$$\frac{\mathrm{1}}{{z}^{\mathrm{2}} \:+\:\mathrm{1}}\:−\:\frac{{iz}}{{z}^{\mathrm{4}} \:−\:\mathrm{1}}\:+\:\frac{{iz}^{\mathrm{3}} }{{z}^{\mathrm{8}} \:−\:\mathrm{1}}\:+\:\frac{{iz}^{\mathrm{7}} }{{z}^{\mathrm{16}} \:−\:\mathrm{1}}\:=\:\mathrm{0}. \\ $$ Answered by ajfour last updated…
Question Number 150527 by Jamshidbek last updated on 13/Aug/21 $$\sqrt[{\mathrm{3}}]{\mathrm{x}\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}−\mathrm{9}\right)−\mathrm{8}}=\mathrm{2x}+\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}} \\ $$ Answered by MJS_new last updated on 13/Aug/21 $$\mathrm{trying}\:\mathrm{something}\:\mathrm{weird}: \\ $$$$\mathrm{assuming}\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} −\mathrm{3}{x}}={y}\in\mathbb{R} \\…
Question Number 84988 by jagoll last updated on 18/Mar/20 $$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{y}\:=\:\mathrm{tan}\:^{\mathrm{2}} \mathrm{z}+\mathrm{cot}\:^{\mathrm{2}} \mathrm{z} \\ $$ Commented by MJS last updated on 18/Mar/20…