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Question Number 144905 Answers: 0 Comments: 1
Question Number 144877 Answers: 1 Comments: 0
$$\:\:\mathrm{u}+\sqrt{\mathrm{u}}+\sqrt[{\mathrm{3}}]{\mathrm{u}}+\sqrt[{\mathrm{4}}]{\mathrm{u}}+\sqrt[{\mathrm{5}}]{\mathrm{u}}+...\:+\infty=? \\ $$$$ \\ $$
Question Number 144872 Answers: 0 Comments: 0
Question Number 144869 Answers: 1 Comments: 0
$${if}\:\:\mathrm{3}^{\boldsymbol{{z}}} \:=\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{5}\sqrt{\mathrm{3}}} \:\centerdot\:\mathrm{3}^{\mathrm{2}} }\:\:{find}\:\:\boldsymbol{{z}}=? \\ $$
Question Number 144901 Answers: 0 Comments: 0
$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{b}^{\mathrm{2}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{\mathrm{3}} }{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}^{\mathrm{3}} }{{b}^{\mathrm{2}} +\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0},\:{n}\:\in\:\mathbb{Z}^{+} \:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{{n}−\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}−\mathrm{1}} }{{b}^{{n}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{{n}+\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}+\mathrm{1}} }{{b}^{{n}} +\mathrm{1}} \\ $$$$ \\ $$
Question Number 144860 Answers: 0 Comments: 2
$$\mid\frac{{x}}{{x}-\mathrm{1}}\mid\:+\:\mid{x}\mid\:=\:\frac{{x}^{\mathrm{2}} }{\mid{x}-\mathrm{1}\mid}\:\:\:{find}\:\:{x}=? \\ $$
Question Number 144858 Answers: 1 Comments: 0
Question Number 144857 Answers: 0 Comments: 0
Question Number 144849 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\mathrm{2}} \frac{\mathrm{1}}{{e}^{\left\{{x}\right\}^{\mathrm{2}} } +\mathrm{1}}{dx}\:\:\:\left\{{x}\right\}\:\:{is}\:{fractional}\:{part}\:{of}\:{x} \\ $$
Question Number 144876 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{dx}=\mathrm{ln}\left(\mathrm{2tan}\:\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$
Question Number 144875 Answers: 1 Comments: 0
Question Number 144839 Answers: 3 Comments: 0
$${If}\:\:{x}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{5}}\:+\:\mathrm{3}\:\:{and}\:\:{y}\:=\:\mathrm{4}\:\sqrt[{\mathrm{3}}]{\mathrm{3}} \\ $$$${Prove}\:{that}:\:\:{x}\:-\:{y}\:<\:\mathrm{0} \\ $$
Question Number 144833 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\alpha\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{4}}{\mathrm{sin}\:\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}=\alpha\: \\ $$$$\mathrm{has}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{0}\:\frac{\Pi}{\mathrm{2}}\right) \\ $$
Question Number 144831 Answers: 2 Comments: 0
$$\frac{\mathrm{3}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}\:+\:\frac{\mathrm{5}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}\:+\:\frac{\mathrm{7}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}\:+\:\frac{\mathrm{9}}{\mathrm{4}\centerdot\mathrm{5}\centerdot\mathrm{6}}\:+\:...\:\infty=? \\ $$
Question Number 144829 Answers: 2 Comments: 0
$$\mathrm{sin}\:^{\mathrm{3}} {x}+\mathrm{cos}\:^{\mathrm{3}} {x}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{2}} {x} \\ $$$${x}=? \\ $$
Question Number 144828 Answers: 2 Comments: 0
$$\mathrm{2sin}\:\mathrm{17}{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{5}{x}+\mathrm{sin}\:\mathrm{5}{x}=\mathrm{0} \\ $$$${x}=? \\ $$
Question Number 144826 Answers: 1 Comments: 0
Question Number 144825 Answers: 1 Comments: 0
$$\underset{\boldsymbol{{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{k}}{{k}^{\mathrm{4}} \:+\:\mathrm{4}}\:=\:? \\ $$
Question Number 144823 Answers: 1 Comments: 0
$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}+\frac{{b}+\mathrm{1}}{{a}+\mathrm{1}}\:\leqslant\:{a}+{b} \\ $$
Question Number 144822 Answers: 1 Comments: 0
$$\mathrm{sin}\:^{\mathrm{3}} {x}\mathrm{cos}\:{x}−\mathrm{cos}\:^{\mathrm{3}} {x}\mathrm{sin}\:{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${x}=? \\ $$
Question Number 144821 Answers: 0 Comments: 1
$$\mathrm{tan}\:\mathrm{193}={k} \\ $$$$\mathrm{cos}\:\mathrm{167}=? \\ $$
Question Number 144820 Answers: 1 Comments: 0
Question Number 144816 Answers: 2 Comments: 0
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{6x}^{\mathrm{2}} }−\left(\mathrm{1}+\mathrm{7x}\right)}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}−\mathrm{3}\right)}\:=? \\ $$
Question Number 144815 Answers: 1 Comments: 0
Question Number 144811 Answers: 0 Comments: 0
$$\int\left\{\frac{\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} −{tan}^{\mathrm{2}} {x}}}\right\}{dx} \\ $$
Question Number 144810 Answers: 0 Comments: 0
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