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Question Number 33129 Answers: 0 Comments: 2
$$\left.\mathrm{1}\right){find}\:{the}\:{value}\:{of}\:\:\:{u}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{4}\:+{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:. \\ $$
Question Number 33128 Answers: 0 Comments: 2
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$
Question Number 33127 Answers: 0 Comments: 1
$$\:{find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:\:{with}\:\mathrm{0}<{a}<\pi\:. \\ $$
Question Number 33126 Answers: 0 Comments: 1
$${let}\:{give}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:−\mathrm{3}{x}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{if}\:\:\:\:{f}\left({x}\right)=\Sigma\:{a}_{{n}} \:{x}^{{n}} \:\:{calculate}\:{the}\:{sequence}\:{a}_{{n}} \\ $$
Question Number 33122 Answers: 0 Comments: 0
$${let}\:{considere}\:{f}\:{and}\:{u}\:{differenciable}\:{function}\:{prove} \\ $$$${that}\:\frac{{d}}{{dt}}\left(\:\int_{{a}} ^{{u}\left({t}\right)} {f}\left({t},{x}\right){dx}\right)=\int_{{a}} ^{{u}\left({t}\right)} \:\frac{\partial{f}}{\partial{t}}\left({t},{x}\right){dx}\:+{f}\left({t},{u}\left({t}\right)\right){u}^{'} \left({t}\right) \\ $$
Question Number 33120 Answers: 1 Comments: 0
$${let}\:{give}\:\alpha>\mathrm{0}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+\alpha{x}\right)}}\:. \\ $$
Question Number 33119 Answers: 0 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{n}} }{{e}^{{t}} \:−\mathrm{1}}\:{dt}\:{by}\:{using}\:\xi\left({x}\right)\:{for}\:{n}\:{integr} \\ $$$$\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{with}\:{x}>\mathrm{1}\:. \\ $$
Question Number 33116 Answers: 1 Comments: 0
$${solve}\:{at}\:\left[\mathrm{0},\pi\right]\:\:{cos}\alpha\:+{cos}\left(\mathrm{2}\alpha\right)\:+{cos}\left(\mathrm{3}\alpha\right)=\mathrm{0} \\ $$
Question Number 33134 Answers: 0 Comments: 0
$${describe}\:{geometrically}\:{the}\:{transformation} \\ $$$${with}\:{matrix} \\ $$$$\left.\mathrm{1}\left.\right)\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}}\end{pmatrix}\:\:\:\mathrm{2}\right)\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$
Question Number 33103 Answers: 1 Comments: 1
Question Number 33100 Answers: 1 Comments: 0
Question Number 33135 Answers: 0 Comments: 0
$${the}\:{triangle}\:{with}\:{vertices}\:{A}\left(−\mathrm{1},−\mathrm{3}\right) \\ $$$$,{B}\left(\mathrm{2},\mathrm{1}\right),{and}\:{C}\left(−\mathrm{2},\mathrm{2}\right),\:{is}\:{transformed} \\ $$$${by}\:{matrix}\:\begin{pmatrix}{{a}\:\:\:\:\:\:{b}}\\{{c}\:\:\:\:\:\:\:{d}}\end{pmatrix}\:{into}\:{the}\:{triangle}\:{with}\:{vertices} \\ $$$${A}\left(−\mathrm{2},−\mathrm{3}\bar {\right)},\:{B}\left(\mathrm{4},\mathrm{1}\right),{C}\left(−\bar {\mathrm{4}},\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\: \\ $$$${a},{b},{c}\:{and}\:{d} \\ $$$$ \\ $$
Question Number 33096 Answers: 1 Comments: 0
$${z}\:{and}\:{w}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid\mid{z}\mid−\mid{w}\mid\mid\:\leqslant\:\mid{z}−{w}\mid\:{and}\:\mid{z}\mid−\mid{w}\mid\leqslant\:\mid{z}+{w}\mid \\ $$
Question Number 33095 Answers: 1 Comments: 0
$${z}={x}+{yi}\:\in\:\mathbb{C} \\ $$$$\overset{−} {{z}}={x}−{y}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid{z}\mid^{\mathrm{2}} =\mid{z}^{\mathrm{2}} \mid={z}\overset{−} {{z}},\:{so}\:{z}\neq\mathrm{0}\:\rightarrow\frac{\mathrm{1}}{{z}}=\frac{\overset{−} {{z}}}{\mid{z}\mid^{\mathrm{2}} } \\ $$
Question Number 33094 Answers: 0 Comments: 1
$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:\:{dvelopp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$
Question Number 33090 Answers: 1 Comments: 0
Question Number 33089 Answers: 1 Comments: 5
$$\:\boldsymbol{\mathrm{T}}\mathrm{he}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{GCF}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\:\mathrm{360}\:\boldsymbol{\mathrm{and}}\:\mathrm{6}\:\boldsymbol{\mathrm{respectively}}.\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}} \\ $$$$\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{are}}\:\mathrm{18}\:\boldsymbol{\mathrm{and}}\:\mathrm{60}. \\ $$$$\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{number}}. \\ $$
Question Number 33088 Answers: 0 Comments: 3
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$
Question Number 33074 Answers: 1 Comments: 1
$${find}\:{interms}\:{of}\:{n}\:\:{the}\:{sum}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:\:{C}_{{n}} ^{{k}} \\ $$
Question Number 33073 Answers: 2 Comments: 1
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} {e}^{−{x}} \\ $$
Question Number 33072 Answers: 1 Comments: 1
$${find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{{n}} ^{{k}} \:. \\ $$
Question Number 33069 Answers: 0 Comments: 0
$${by}\:\:{using}\:{residus}\:{theorem}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}\:{dt}\:=\:\frac{\pi}{{sin}\left(\pi{a}\right)}\:{with}\:\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$
Question Number 33064 Answers: 1 Comments: 2
Question Number 33153 Answers: 0 Comments: 1
$${it}\:{is}\:{given}\:{that} \\ $$$$\:\:\underset{{r}=\mathrm{1}\:} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]=\mathrm{200} \\ $$$${and} \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]^{\mathrm{2}} =\mathrm{2800} \\ $$$${find}\:{the}\:{value}\:{of} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)\right]^{\mathrm{2}} \\ $$$$ \\ $$
Question Number 33052 Answers: 0 Comments: 1
Question Number 33051 Answers: 0 Comments: 3
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