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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
Question Number 144801 Answers: 1 Comments: 2
$${if}\:{you}\:{know}\:{that}\:{the}\:{probability}\:{of}\:{a}\:{picture} \\ $$$${appearing}\:{when}\:{acoin}\:{is}\:{tossed}\:{is}\:\mathrm{2}/\mathrm{5} \\ $$$${then}\:{the}\:{probability}\:{of}\:{getting}\:{writings}\: \\ $$$${when}\:{this}\:{coin}\:{is}\:{tossed}\:\mathrm{6}\:{times}\:? \\ $$
Question Number 144800 Answers: 1 Comments: 0
$$\mathrm{Let}\:\beta\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{angle}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{4xcos}\:\beta+\mathrm{cot}\:\beta=\mathrm{0} \\ $$$$\mathrm{involving}\:\mathrm{variable}\:\mathrm{x}\:\mathrm{has}\:\mathrm{multiple} \\ $$$$\mathrm{roots}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{measure}\:\mathrm{of}\:\beta\:\mathrm{in} \\ $$$$\mathrm{radians}\:\mathrm{is}\:\_\_ \\ $$
Question Number 144799 Answers: 0 Comments: 0
Question Number 144794 Answers: 0 Comments: 0
Question Number 144792 Answers: 1 Comments: 0
$$\:\int\:\frac{\mathrm{2x}^{\mathrm{3}} −\mathrm{1}}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}}\:\mathrm{dx}\:? \\ $$
Question Number 144791 Answers: 1 Comments: 0
$$\:\mathrm{Express}\:\mathrm{sin}\:\mathrm{5x}\:\mathrm{as}\:\mathrm{polynomial} \\ $$$$\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{sin}\:\mathrm{x}.\: \\ $$
Question Number 144813 Answers: 0 Comments: 1
$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\:\left({x}\right)}{\mathrm{1}\:+\:{x}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left({x}\:\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\:{n}} \:{x}^{\:\mathrm{2}{n}} \:{dx} \\ $$$$\:\:\:\:\:\:\:\::=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\left(\:−\mathrm{1}\:\right)^{\:{n}} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\:\mathrm{2}{n}} \:\mathrm{ln}\left(\:{x}\:\right){dx} \\ $$$$\:\:\:\:\:\:\:\::\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\:−\mathrm{1}\:\right)^{\:{n}} \left\{\:\left[\frac{{x}^{\:\mathrm{2}{n}+\mathrm{1}} }{\mathrm{2}{n}\:+\mathrm{1}}\:\mathrm{ln}\:\left(\:{x}\:\right)\right]_{\mathrm{0}} ^{\:\mathrm{1}} −\frac{\mathrm{1}}{\left(\mathrm{2}{n}\:+\mathrm{1}\:\right)^{\:\mathrm{2}} \:}\:\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\::\:=\:\underset{{n}=\mathrm{1}} {\overset{\:\infty} {\sum}}\frac{\left(\:−\mathrm{1}\:\right)^{\:{n}−\mathrm{1}} }{\left(\:\mathrm{2}{n}\:+\mathrm{1}\right)^{\:\mathrm{2}} }\:=\:−\mathrm{G}\:\:\left(\mathrm{Catalan}\:\mathrm{constant}\:\right) \\ $$
Question Number 144789 Answers: 0 Comments: 0
Question Number 144788 Answers: 1 Comments: 0
Question Number 144787 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Q}\:::\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:#\:\mathrm{Calculus}\:# \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{If}\::\:\:\:\:\:\:\:\:\boldsymbol{\phi}\:\left(\:{n}\:\right)\::\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{x}^{\:\mathrm{2}{n}} }{\mathrm{1}\:+\:{x}^{\:\mathrm{2}} }\:\mathrm{d}{x}\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{then}\:\:\mathrm{find}\:\:\mathrm{the}\:\:\mathrm{value}\:\mathrm{of}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{S}\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(\:−\mathrm{1}\:\right)^{\:{n}} \:\boldsymbol{\phi}\:\left(\:{n}\:\right)}{{n}}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{m}.\mathrm{n}.\mathrm{july}.\mathrm{1970} \\ $$
Question Number 144785 Answers: 0 Comments: 0
Question Number 144783 Answers: 0 Comments: 1
Question Number 144782 Answers: 0 Comments: 0
Question Number 144781 Answers: 1 Comments: 0
Question Number 144780 Answers: 0 Comments: 0
$$\:\mathrm{Let}\:\mathrm{x}\in\left[−\frac{\mathrm{5}\pi}{\mathrm{12}},−\frac{\pi}{\mathrm{3}}\right]\:,\:\mathrm{then}\:\mathrm{the}\: \\ $$$$\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{y}=\mathrm{tan}\:\left(\mathrm{x}+\frac{\mathrm{2}\pi}{\mathrm{3}}\right)−\mathrm{tan}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{6}}\right)+\mathrm{cos}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{6}}\right) \\ $$$$\:\mathrm{is}\: \\ $$
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