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Question Number 156992 Answers: 1 Comments: 1
Question Number 156991 Answers: 1 Comments: 0
Question Number 156979 Answers: 2 Comments: 2
Question Number 156977 Answers: 0 Comments: 0
$$\:\begin{cases}{{a}_{\mathrm{1}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\\{{a}_{{n}+\mathrm{1}} =\mathrm{4}{a}_{{n}} ^{\mathrm{3}} −\mathrm{3}{a}_{{n}} \:;\:\forall{n}\geqslant\mathrm{1}}\end{cases} \\ $$$$\:{a}_{{n}} =? \\ $$
Question Number 156973 Answers: 0 Comments: 1
Question Number 156993 Answers: 0 Comments: 3
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{n}} {\underbrace{\left({sin}\left({sin}\left({sin}\ldots\left({sin}\left({x}\right)\right)\ldots\right)}}\:\sqrt{{n}}=?\right.\right. \\ $$$$\mathrm{0}<{x}<\pi \\ $$
Question Number 157003 Answers: 1 Comments: 0
$$\mathrm{let}\:\:\boldsymbol{\mathrm{n}}\in\mathbb{Z}^{+} \\ $$$$\mathrm{shov}\:\mathrm{that}\:\:\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{sin}\left(\mathrm{x}^{-\boldsymbol{\mathrm{n}}} \right)\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{dx}\:=\:\frac{\pi\boldsymbol{\gamma}}{\mathrm{2n}^{\mathrm{2}} }\: \\ $$$$\mathrm{where}\:\:\boldsymbol{\gamma}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{Euler}-\mathrm{Mascheroni}\:\mathrm{constan}\: \\ $$
Question Number 156969 Answers: 0 Comments: 0
Question Number 156968 Answers: 0 Comments: 0
Question Number 156966 Answers: 2 Comments: 1
Question Number 156962 Answers: 1 Comments: 0
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{3}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}\:+\:\mathrm{x}}}\:+\:\frac{\mathrm{x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }}\:=\:\mathrm{2}\:\sqrt[{\mathrm{3}}]{\mathrm{4}} \\ $$
Question Number 156961 Answers: 0 Comments: 0
$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)\:-\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{3}} }\:\mathrm{dx}\:=\:? \\ $$
Question Number 156951 Answers: 1 Comments: 7
$${solve}\:{for}\:{n}\in{N} \\ $$$$\left({n}−\mathrm{1}\right)!+\mathrm{1}={n}^{\mathrm{2}} \\ $$
Question Number 156942 Answers: 0 Comments: 0
Question Number 156940 Answers: 1 Comments: 0
Question Number 156933 Answers: 1 Comments: 0
Question Number 156931 Answers: 1 Comments: 2
$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}''−\mathrm{4}{xy}'−\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}=\mathrm{0} \\ $$
Question Number 156927 Answers: 0 Comments: 0
$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{\left(\mathrm{1}\:-\:\mathrm{x}\right)\centerdot\mathrm{x}^{\boldsymbol{\mathrm{n}}+\mathrm{4}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}\boldsymbol{\mathrm{n}}} }\:\mathrm{dx}\:=\:? \\ $$
Question Number 156924 Answers: 1 Comments: 0
$$\:\underset{{k}=\mathrm{2}} {\overset{\infty} {\prod}}\:\left(\frac{{k}^{\mathrm{3}} −\mathrm{1}}{{k}^{\mathrm{3}} +\mathrm{1}}\right)\:=? \\ $$
Question Number 156923 Answers: 1 Comments: 0
$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\mathrm{x}\:\mathrm{log}\:\left(\mathrm{1}\:+\:\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\:\mathrm{dx}\:=\:? \\ $$
Question Number 156921 Answers: 1 Comments: 0
$${x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0} \\ $$$$\:{x}\in\mathbb{R} \\ $$
Question Number 156913 Answers: 1 Comments: 0
Question Number 157054 Answers: 0 Comments: 1
$${suppose}\:{you}\:{drop}\:{a}\:{tennis}\:{ball}\:{from}\:{a}\:{hieght}\:{of}\:\mathrm{15}\:{feet}.{after}\:{the}\:{ballhits}\:{the}\:{floor}\:{it}\:{rebounds}\:\:{to}\mathrm{85\%}\:{of}\:{its}\:{previous}\:{height}.{how}\:{high}\:{will}\:{the}\:{ball}\:{rebound}\:{after}\:{its}\:{ghird}\:{bounce}\:{round}\:{tl}\:{the}\:{nearest}\:{tenth} \\ $$$$ \\ $$
Question Number 156911 Answers: 0 Comments: 0
$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\left(\left[\mathrm{nx}\right]\:\centerdot\:\mid\mathrm{x}\:-\:\left[\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mid\right]\right)\mathrm{dx} \\ $$$$\left[\ast\right]\:-\:\mathrm{GIF} \\ $$
Question Number 156910 Answers: 0 Comments: 2
Question Number 156909 Answers: 0 Comments: 0
$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{n}\:-\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\left(\mathrm{e}\:-\:\mathrm{1}\right)\centerdot\mathrm{n}}{\mathrm{n}\:+\:\left(\mathrm{e}\:-\:\mathrm{1}\right)\centerdot\mathrm{k}}\right)\:=\:? \\ $$
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