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Question Number 41706 Answers: 1 Comments: 1
$${study}\:{the}\:{convergence}\:{of}\: \\ $$$${u}_{{n}} =\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\mathrm{1}}{{kln}\left({k}\right)}\:−{ln}\left({ln}\left({n}\right)\right. \\ $$
Question Number 41705 Answers: 0 Comments: 1
$${find}\:{nsture}\:{of}\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$$${with}\:{u}_{{n}} =^{{n}} \sqrt{\frac{{n}}{{n}+\mathrm{1}}}\:\:\:−\mathrm{1} \\ $$
Question Number 41704 Answers: 0 Comments: 0
$${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\sqrt{{n}−\mathrm{1}}}\:+\frac{\mathrm{2}}{\sqrt{{n}}}\:+\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}} \\ $$$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:{u}_{{n}} \\ $$
Question Number 41703 Answers: 0 Comments: 1
$${calculate}\:\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{cosx}}{{cos}^{\mathrm{3}} {x}\:+{sin}^{\mathrm{3}} {x}}{dx} \\ $$
Question Number 41702 Answers: 1 Comments: 3
$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{arcsin}\left(\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dt} \\ $$
Question Number 41699 Answers: 0 Comments: 0
$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{perpendicular}}\:\boldsymbol{\mathrm{distance}}\:\boldsymbol{\mathrm{btn}} \\ $$$$\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{lines}}\:\boldsymbol{{y}}=\mathrm{2}\boldsymbol{{x}}−\mathrm{3} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{{y}}=\mathrm{3}\boldsymbol{{x}}−\mathrm{1} \\ $$
Question Number 44610 Answers: 0 Comments: 1
$${Let}\:{A}\:{be}\:\mathrm{2}×\mathrm{3}\:{matrix}\:,\:{whereas}\:{B}\:{be} \\ $$$$\mathrm{3}×\mathrm{2}\:{matrix}.\:{If}\:{determinant}\left({AB}\right)=\mathrm{4}, \\ $$$${then}\:{the}\:{value}\:{of}\:{determinant}\:\left({BA}\right)\:? \\ $$
Question Number 41691 Answers: 2 Comments: 0
$${tan}^{\mathrm{2}} \mathrm{20}+{tan}^{\mathrm{2}} \mathrm{40}+{tan}^{\mathrm{2}} \mathrm{80}=\mathrm{33} \\ $$
Question Number 41686 Answers: 1 Comments: 2
Question Number 41682 Answers: 2 Comments: 0
$${find}\:{radius}\:{of}\:{curvature}\:{to} \\ $$$${y}=\mathrm{sin}\:{x}\:\:{at}\:\:{x}=\pi/\mathrm{6}\:. \\ $$
Question Number 41679 Answers: 1 Comments: 5
$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{t}\:+{xt}^{\mathrm{2}} \right){dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:{then}\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{t}\:+{t}^{\mathrm{2}} \right){dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}−{t}^{\mathrm{3}} \right){dt}\:. \\ $$
Question Number 41678 Answers: 1 Comments: 0
$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}=\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\:{by}\:{using} \\ $$$${only}\:{series}. \\ $$
Question Number 41677 Answers: 2 Comments: 2
$${calculate}\:{A}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{\mathrm{8}} {xdx}\:{and}\: \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{\mathrm{8}} {xdx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}\:+{B}\:{and}\:{A}−{B} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{A}^{\mathrm{2}} \:−{B}^{\mathrm{2}} \\ $$
Question Number 41675 Answers: 1 Comments: 1
Question Number 41672 Answers: 0 Comments: 1
$${find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}}\:{interms}\:{of}\:{H}_{{n}} \\ $$
Question Number 41671 Answers: 0 Comments: 0
$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{p}}=\mathrm{6}.\mathrm{4}×\mathrm{10}^{\mathrm{4}} \:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{q}}=\mathrm{3}.\mathrm{2}×\mathrm{10}^{\mathrm{5}} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{values}}\:\boldsymbol{\mathrm{of}} \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\boldsymbol{\mathrm{p}}×\boldsymbol{\mathrm{q}} \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}} \\ $$$$\boldsymbol{\mathrm{write}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{answers}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{standard}}\:\boldsymbol{\mathrm{form}} \\ $$
Question Number 41651 Answers: 2 Comments: 1
$$\int\left(\:\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}^{\mathrm{3}} +.........\right)\:{dx}\:,\:\:\: \\ $$$$\left(\mathrm{0}<\mid{x}\mid<\mathrm{1}\right) \\ $$
Question Number 41642 Answers: 1 Comments: 0
$$\mathrm{n}\left(\mathrm{n}\:−\:\mathrm{1}\right)\left(\mathrm{n}\:−\:\mathrm{2}\right)\left(\mathrm{n}\:−\:\mathrm{3}\right)\:....\:\left(\mathrm{n}\:−\:\mathrm{r}\:+\:\mathrm{1}\right)\:=\:?? \\ $$
Question Number 41634 Answers: 2 Comments: 1
Question Number 41622 Answers: 4 Comments: 9
$${let}\:{z}_{\mathrm{1}} \:{and}\:{z}_{\mathrm{2}} \:{the}\:{roots}\:{of}\:{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{z}_{\mathrm{1}} ^{\mathrm{3}} \:+{z}_{\mathrm{2}} ^{\mathrm{3}} \:\:\:{then}\:\:\frac{\mathrm{1}}{{z}_{\mathrm{1}} ^{\mathrm{3}} }\:+\frac{\mathrm{1}}{{z}_{\mathrm{2}} ^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{z}_{\mathrm{1}} ^{\mathrm{4}} \:+{z}_{\mathrm{2}} ^{\mathrm{4}} \:\:{then}\:\:\frac{\mathrm{1}}{{z}_{\mathrm{1}} ^{\mathrm{4}} }\:+\frac{\mathrm{1}}{{z}_{\mathrm{2}} ^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right)\:{let}\:{n}\:{from}\:{N}\:\:{simplify} \\ $$$${A}_{{n}} =\:{z}_{\mathrm{1}} ^{{n}} \:+{z}_{\mathrm{2}} ^{{n}} \:\:\:\:\:\:\:{and}\:\:{B}_{{n}} =\:{z}_{\mathrm{1}} ^{{n}} \:−{z}_{\mathrm{2}} ^{{n}} \\ $$$$\left.\mathrm{4}\right)\:{simplify}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left({z}_{\mathrm{1}} ^{{k}} \:\:\:+{z}_{\mathrm{2}} ^{{k}} \right) \\ $$
Question Number 41620 Answers: 2 Comments: 1
Question Number 41616 Answers: 0 Comments: 0
Question Number 41627 Answers: 1 Comments: 2
Question Number 41787 Answers: 1 Comments: 0
Question Number 41783 Answers: 1 Comments: 0
Question Number 41606 Answers: 2 Comments: 0
$$\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\mathrm{16}\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{24}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{9}\boldsymbol{\mathrm{x}}−\mathrm{1}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{method}}.\:\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{satisfy}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{polynomial}} \\ $$
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