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Question Number 115401 Answers: 2 Comments: 0
$$\int\frac{\sqrt{\mathrm{x}}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx}=? \\ $$
Question Number 115391 Answers: 1 Comments: 0
$$\mathrm{Who}\:\mathrm{can}\:\mathrm{explain} \\ $$$$\mathrm{me}\:\mathrm{what}\:\mathrm{is}\:``\mathbb{R}_{{n}} \left[{x}\right]''? \\ $$$${In}\:{French}\:{if}\:{possible}. \\ $$
Question Number 115387 Answers: 0 Comments: 6
Question Number 115384 Answers: 1 Comments: 0
$${solve}\:{the}\:{limit}\:{problem} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{cos}{x}}{\mathrm{1}−{e}^{−{x}} }\right) \\ $$
Question Number 115471 Answers: 0 Comments: 1
$${For}\:{angles}\:{a},{b},{c}\in\mathbb{R}\:{with}\:{a}+{b}+{c}=\pi,\: \\ $$$${prove}\:{the}\:{following}\:{identities}: \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{{a}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{c}}{\mathrm{2}}\right)=\left(\mathrm{tan}\left(\frac{{a}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{b}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{c}}{\mathrm{2}}\right)\right)^{−\mathrm{1}} \\ $$$$\mathrm{H}{elp} \\ $$$$ \\ $$
Question Number 115380 Answers: 2 Comments: 0
$${if}\: \\ $$$${a}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{3}{b}^{\mathrm{2}} =\mathrm{1},{prove}\:{that}\: \\ $$$$\left({a}+\mathrm{3}{b}\right)^{\mathrm{3}} \frac{{d}^{\mathrm{2}} {b}}{{da}^{\mathrm{2}} }+\mathrm{2}\left({a}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{3}{b}^{\mathrm{2}} \right)=\mathrm{0} \\ $$
Question Number 115368 Answers: 1 Comments: 0
$${an}\:{open}\:{rectanqular}\:{container}\:{is}\:{to} \\ $$$${have}\:{a}\:{volume}\:{of}\:\mathrm{62}.\mathrm{5}{cm}^{\mathrm{3}} .{find}\:{the}\:{least} \\ $$$${possible}\:{surface}\:{area}\:{of}\:{the}\:{material} \\ $$$${required} \\ $$
Question Number 115367 Answers: 1 Comments: 0
$${solve}\:{xy}^{''} −\left({x}^{\mathrm{2}} +\mathrm{1}\right){y}^{'} \:\:={x}^{\mathrm{2}} {sin}\left(\mathrm{2}{x}\right) \\ $$
Question Number 115366 Answers: 2 Comments: 0
$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{2}} \:\frac{{dx}}{{ch}^{\mathrm{2}} {x}\:+{sh}^{\mathrm{2}} {x}} \\ $$
Question Number 115365 Answers: 0 Comments: 0
$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\frac{{arctan}\left({xy}\right)}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }}{dxdy} \\ $$
Question Number 115364 Answers: 1 Comments: 0
$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\sqrt{{xy}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$
Question Number 115363 Answers: 1 Comments: 0
$${evaluate} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{cos}^{\mathrm{2}} {x}}{dx} \\ $$
Question Number 115362 Answers: 1 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{sinx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$
Question Number 115361 Answers: 1 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$
Question Number 115348 Answers: 1 Comments: 0
$${If}\:{x}\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:{and}\:\mathrm{2cos}\:{x}\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)+\mathrm{tan}\:^{\mathrm{2}} {x}\:<\:\mathrm{sec}\:^{\mathrm{2}} {x}\: \\ $$$${has}\:{solution}\:{set}\:{is}\:{a}<{x}<{b}.\:{find}\:{the} \\ $$$${value}\:{of}\:{a}+{b} \\ $$
Question Number 115345 Answers: 3 Comments: 0
$$\mathrm{sec}\:\theta\:\left(\mathrm{sec}\:\theta\:\left(\mathrm{sin}\:^{\mathrm{2}} \theta\right)+\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right)=\mathrm{1} \\ $$$${has}\:{the}\:{roots}\:{are}\:\theta_{\mathrm{1}} \:{and}\:\theta_{\mathrm{2}} .\:{Find}\:{the} \\ $$$${value}\:{of}\:\mathrm{tan}\:\theta_{\mathrm{1}} ×\mathrm{tan}\:\theta_{\mathrm{2}} . \\ $$
Question Number 115341 Answers: 1 Comments: 0
$${If}\:\mathrm{log}\:\mathrm{tan}\:\mathrm{1}°+\mathrm{log}\:\mathrm{tan}\:\mathrm{2}°+\mathrm{log}\:\mathrm{tan}\:\mathrm{3}°+...+\mathrm{log}\:\mathrm{tan}\:\mathrm{89}°={p} \\ $$$${then}\:{p}^{\mathrm{2}} +\mathrm{3}\:=\: \\ $$
Question Number 115333 Answers: 1 Comments: 3
Question Number 115332 Answers: 4 Comments: 3
$${Minimum}\:{value}\:{of}\:{function}\: \\ $$$${f}\left({x}\right)=\:\frac{\mathrm{16}{x}^{\mathrm{2}} \:\mathrm{cos}\:^{\mathrm{2}} {x}+\mathrm{4}}{{x}\:\mathrm{cos}\:{x}}\:{where}\:−\pi<{x}<\mathrm{0} \\ $$
Question Number 115328 Answers: 1 Comments: 0
$${If}\:\frac{\mathrm{sin}\:\mathrm{1}°+\mathrm{sin}\:\mathrm{2}°+\mathrm{sin}\:\mathrm{3}°+...+\mathrm{sin}\:\mathrm{44}°}{\mathrm{cos}\:\mathrm{1}°+\mathrm{cos}\:\mathrm{2}°+\mathrm{cos}\:\mathrm{3}°+...+\mathrm{cos}\:\mathrm{44}°}=\chi \\ $$$${then}\:\chi^{\mathrm{4}} +\mathrm{4}\chi^{\mathrm{3}} +\mathrm{4}\chi^{\mathrm{2}} +\mathrm{4}= \\ $$
Question Number 115325 Answers: 0 Comments: 5
Question Number 115320 Answers: 4 Comments: 0
$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:^{\mathrm{6}} \left(\mathrm{2}{x}\right)\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{3}{x}\right)}{\mathrm{3}{x}^{\mathrm{2}} }\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{4}{x}+\mathrm{2sin}\:^{\mathrm{2}} {x}.\mathrm{cos}\:\mathrm{4}{x}}{{x}^{\mathrm{2}} .\mathrm{cos}\:\mathrm{3}{x}}? \\ $$$$\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}−\mathrm{2cos}\:^{\mathrm{2}} {x}−\mathrm{1}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}}−\sqrt{\mathrm{sin}\:{x}}}\:?\: \\ $$
Question Number 115318 Answers: 2 Comments: 0
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\mathrm{sin}\:{x}}{\mathrm{2sin}\:^{\mathrm{2}} \left(\mathrm{3}{x}\right)−{x}^{\mathrm{2}} \mathrm{cos}\:{x}} \\ $$
Question Number 115312 Answers: 3 Comments: 2
$${Solve}\:\:{for}\:\:{x}\:\in\:\mathbb{R}\:\:{that}\:\:{suitable}\:\:{on}\:\:{this} \\ $$$${inequality}\::\:\:\:\:\sqrt{\mathrm{8}−{x}^{\mathrm{2}} }\:\:>\:\:{x} \\ $$
Question Number 115302 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:...\:{nice}\:\:{math}... \\ $$$$\:\:\:\:\:{find} \\ $$$$\:\:\:\:{lim}_{{n}\rightarrow\infty\:\:} \left\{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\int_{{k}−\mathrm{1}} ^{\:\:{k}} {tan}^{−\mathrm{1}} \left(\frac{{nx}−{nk}}{{kx}+{n}^{\mathrm{2}} }\right){dx}\right\} \\ $$$$\: \\ $$
Question Number 115301 Answers: 2 Comments: 0
$${if}\: \\ $$$${jx}^{\mathrm{2}} +\mathrm{2}{kxy}+{by}^{\mathrm{2}} =\mathrm{1}\:{show}\:{that} \\ $$$$\left({kx}+{by}\right)^{\mathrm{3}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }={k}^{\mathrm{2}} −{jb} \\ $$
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