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Question Number 141372 Answers: 3 Comments: 0
Question Number 141368 Answers: 2 Comments: 0
$${A}\:{closed}\:{cylindrical}\:{can}\:{be}\:{is}\:{to}\:{hold} \\ $$$$\mathrm{1}\:{liters}\:{of}\:{liquid}\:.\:{How}\:{should}\:{we}\: \\ $$$${choose}\:{the}\:{height}\:{and}\:{radius}\: \\ $$$${to}\:{minimize}\:{the}\:{amount}\:{of} \\ $$$${material}\:{needed}\:{to}\:{manufacture} \\ $$$${the}\:{can}\:?\: \\ $$
Question Number 141367 Answers: 1 Comments: 0
$$\int\left(\sqrt{{cosx}\centerdot{senx}}\right){dx} \\ $$
Question Number 141311 Answers: 0 Comments: 0
$$\mathrm{Show}\:\mathrm{that}\:,\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} +\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}−\mathrm{1}} =\mathrm{C}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{n}−\mathrm{k}} \\ $$
Question Number 141308 Answers: 1 Comments: 0
Question Number 141303 Answers: 3 Comments: 1
Question Number 141304 Answers: 0 Comments: 1
Question Number 141388 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \sqrt{\left({senx}\centerdot{cosx}\right)}{dx} \\ $$$${Help} \\ $$
Question Number 141387 Answers: 1 Comments: 0
$$\int_{−\pi/\mathrm{4}} ^{\pi/\mathrm{4}} \left({sec}^{\mathrm{2}} {x}+{tgx}\right)^{\mathrm{2}} {dx} \\ $$
Question Number 141380 Answers: 0 Comments: 0
Question Number 141378 Answers: 2 Comments: 0
$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\frac{\left(\mathrm{5}{n}+\mathrm{2}\right)\left(\mathrm{5}{n}+\mathrm{3}\right)}{\left(\mathrm{5}{n}+\mathrm{1}\right)\left(\mathrm{5}{n}+\mathrm{4}\right)}\:=\varphi\: \\ $$$$\:\:\:\:\:\:\:\varphi:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$
Question Number 141294 Answers: 5 Comments: 0
$${Find}\:{max}\:\&\:{min}\:{value}\:{of} \\ $$$$\:{f}\left({x}\right)=\frac{{x}}{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{9}}. \\ $$
Question Number 141328 Answers: 2 Comments: 0
$$......\:{Evaluate}: \\ $$$$\:\:\:\:\:\mathscr{F}\::=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{{n}+\mathrm{1}}\:=? \\ $$$$....... \\ $$
Question Number 141312 Answers: 0 Comments: 0
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{{Si}\left(\mathrm{2}\pi{n}\right)−\frac{\pi}{\mathrm{2}}}{{n}}=? \\ $$
Question Number 141289 Answers: 1 Comments: 1
Question Number 141381 Answers: 0 Comments: 0
$$\mathrm{Let}\:\:{a},{b}\:\geqslant\:\mathrm{0}\:.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+{b}+\mathrm{2}\right)^{\mathrm{3}} \:\geqslant\:\frac{\mathrm{27}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 141291 Answers: 0 Comments: 3
Question Number 141269 Answers: 1 Comments: 3
$$\theta+\phi+\psi=\pi\:\:\left({angles}\:{of}\:{a}\:\bigtriangleup\right) \\ $$$${find}\:{maximum}\:{of} \\ $$$$\:\left(\phi−\theta\right)^{\mathrm{2}} +\left(\psi−\phi\right)^{\mathrm{2}} +\left(\psi−\theta\right)^{\mathrm{2}} \:. \\ $$
Question Number 141412 Answers: 1 Comments: 2
$$\:{Find}\:{the}\:{range}\:{of}\:{real}\:{number} \\ $$$${of}\:{q}\:{such}\:{that}\:{the}\:{function}\: \\ $$$$\:{f}\left({x}\right)\:=\:\mathrm{cos}\:{x}\left({q}\:\mathrm{sin}\:^{\mathrm{2}} {x}−\mathrm{5}\right)\:{have} \\ $$$${minimum}\:{value}\:{is}\:−\mathrm{5}\:. \\ $$
Question Number 141283 Answers: 0 Comments: 0
$$\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}^{\mathrm{3}} }−\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{3}^{\mathrm{3}} }+\frac{\zeta\left(\mathrm{4}\right)}{\mathrm{4}^{\mathrm{3}} }−\frac{\zeta\left(\mathrm{5}\right)}{\mathrm{5}^{\mathrm{3}} }+... \\ $$
Question Number 141257 Answers: 0 Comments: 0
Question Number 141252 Answers: 1 Comments: 0
Question Number 141249 Answers: 2 Comments: 1
Question Number 141246 Answers: 1 Comments: 0
$${r}={q}+\mathrm{1} \\ $$$${pq}={q}+\mathrm{1} \\ $$$${c}^{\mathrm{2}} {p}={qr}^{\mathrm{2}} \:\:\:,\:{help}\:{find}\:{p},\:{q},\:{r}. \\ $$
Question Number 141244 Answers: 1 Comments: 1
$${I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{x}\left\{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right){x}−\mathrm{2}{a}^{\mathrm{2}} {x}^{\mathrm{2}} −\mathrm{2}{b}^{\mathrm{2}} \right\}{dx}}{\left({a}^{\mathrm{2}} {x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{2}} \left\{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right){x}+{a}^{\mathrm{2}} {x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right\}} \\ $$
Question Number 141240 Answers: 0 Comments: 0
$$\mathrm{Solve}\:\mathrm{using}\:\mathrm{fourier}'\mathrm{s}\:\mathrm{series} \\ $$$$−\mathrm{y}''+\mathrm{y}=\mathrm{e}^{−\mathrm{2}\mid\mathrm{x}\mid} \\ $$
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