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Relation and FunctionsQuestion and Answers: Page 11
Question Number 143702 Answers: 2 Comments: 0
$${n}\:\in\:\mathrm{IN}. \\ $$$${I}_{{n}} \:=\:\int_{\mathrm{1}} ^{\:\mathrm{e}} {x}^{{n}+\mathrm{1}} {lnx}\:{dx}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\left(\boldsymbol{{I}}_{\boldsymbol{{n}}} \right)\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{increasing}}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{part}}−\boldsymbol{\mathrm{by}}−\boldsymbol{\mathrm{part}}\:\boldsymbol{\mathrm{integration}},\:\boldsymbol{\mathrm{calculate}}\:\boldsymbol{{I}}_{\boldsymbol{{n}}} . \\ $$
Question Number 143606 Answers: 1 Comments: 0
Question Number 143576 Answers: 0 Comments: 0
$${find}\:{L}\left(\frac{{arctanx}}{{x}}\right) \\ $$
Question Number 143575 Answers: 1 Comments: 0
$${find}\:{L}\left({e}^{−\sqrt{{x}}} \right) \\ $$
Question Number 143562 Answers: 0 Comments: 0
Question Number 143546 Answers: 1 Comments: 0
$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{log}^{\mathrm{2}} \mathrm{x}}{\left(\mathrm{8}+\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{2}} }\mathrm{dx} \\ $$
Question Number 143488 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{2}+\mathrm{sinx}} \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$
Question Number 143383 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\frac{\sqrt{{x}}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{\mathrm{2}{x}+\mathrm{3}}}{dx} \\ $$
Question Number 143382 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} {xe}^{−{x}^{\mathrm{2}} } {log}\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$
Question Number 143381 Answers: 2 Comments: 0
$${let}\:{f}\left({x}\right)={arctan}\left(\sqrt{\mathrm{2}}{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right){and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){if}\:{f}\left({x}\right)=\Sigma{a}_{{n}} {x}^{{n}} \:\:{find}\:{the}\: \\ $$$${sequence}\:{a}_{{n}} \\ $$
Question Number 143380 Answers: 1 Comments: 0
$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}} \\ $$$${by}\:{use}\:{of}\:{two}\:{methods} \\ $$
Question Number 143262 Answers: 1 Comments: 0
$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}\:+\mathrm{2}{sinx}} \\ $$
Question Number 143261 Answers: 1 Comments: 0
$${find}\:{Y}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)....\left({x}+{n}\right)} \\ $$$$\left({n}>\mathrm{1}\:{integr}\right) \\ $$
Question Number 143260 Answers: 0 Comments: 1
$${find}\:\int\:\frac{{dx}}{\:\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}+\sqrt{{x}+\mathrm{2}}} \\ $$
Question Number 143259 Answers: 1 Comments: 0
$${solve}\:{y}^{''} −{y}^{'} +\mathrm{2}={xsin}\left(\mathrm{3}{x}\right) \\ $$
Question Number 143258 Answers: 1 Comments: 0
$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{sh}\left({xt}\right)}{{x}+{t}}{dt} \\ $$
Question Number 143257 Answers: 0 Comments: 0
$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} {e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {arctan}\left(\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right){dxdy} \\ $$
Question Number 143256 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{{x}} ^{\mathrm{2}−{x}} {e}^{−{xy}} \sqrt{{x}+{y}}{dy}\:{dx} \\ $$
Question Number 143255 Answers: 2 Comments: 0
$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{sin}\left(\mathrm{1}−{cosx}\right)+\mathrm{1}−{cos}\left({sinx}\right)}{{x}^{\mathrm{2}} } \\ $$
Question Number 143254 Answers: 1 Comments: 0
$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)} \\ $$
Question Number 143253 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}^{\mathrm{3}} \right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$
Question Number 143148 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{v}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{3k}+\mathrm{1}}\:\mathrm{interms}\:\mathrm{of}\:\mathrm{H}_{\mathrm{n}} \\ $$$$\mathrm{H}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{k}} \\ $$
Question Number 143147 Answers: 0 Comments: 1
$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{lasuite}\:\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{H}_{\mathrm{n}} }{\mathrm{n}^{\mathrm{2}} }\:\mathrm{est}\:\mathrm{bornee} \\ $$$$\mathrm{H}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$
Question Number 142988 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}} +\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\frac{\mathrm{2}}{\:\sqrt{\mathrm{n}}} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{equivalent}\:\mathrm{of}\:\mathrm{u}_{\mathrm{n}} \:\:\left(\mathrm{n}\rightarrow\infty\right) \\ $$
Question Number 142987 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} −\lambda\mathrm{u}_{\mathrm{n}−\mathrm{1}} \\ $$$$\lambda\:\mathrm{real} \\ $$
Question Number 142980 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } \mathrm{log}\left(\mathrm{2}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx}\:\:\:\left(\mathrm{n}\geqslant\mathrm{1}\right) \\ $$$$\mathrm{determine}\:\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \:\mathrm{and}\:\Sigma\:\mathrm{nU}_{\mathrm{n}} \\ $$
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