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Question Number 62850    Answers: 1   Comments: 1

Question Number 62844    Answers: 1   Comments: 0

Let p(x) = ax^2 + bx + c be such that p(x) takes real values for real values of x and non−real values for non−real values of x . Prove that a = 0 and find all possible values of c.

$${Let}\:{p}\left({x}\right)\:=\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:\:{be}\:{such}\:{that}\:{p}\left({x}\right)\:{takes}\:{real}\:{values} \\ $$$${for}\:{real}\:{values}\:{of}\:{x}\:{and}\:{non}−{real}\:{values}\:{for}\:{non}−{real} \\ $$$${values}\:{of}\:{x}\:.\:{Prove}\:{that}\:{a}\:=\:\mathrm{0}\:{and}\:{find}\:{all} \\ $$$${possible}\:{values}\:{of}\:{c}. \\ $$

Question Number 62839    Answers: 1   Comments: 3

Question Number 62836    Answers: 0   Comments: 0

Question Number 62833    Answers: 0   Comments: 2

∫((cos(x))/x) dx ∫(√(sin(x) )) dx ∫(√(1−k^2 sin^2 (x))) dx k:constant

$$\int\frac{{cos}\left({x}\right)}{{x}}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{{sin}\left({x}\right)\:}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{\mathrm{1}−{k}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({x}\right)}\:{dx}\:\:\:\:\:\:{k}:{constant} \\ $$

Question Number 62828    Answers: 0   Comments: 1

let U_n =∫_0 ^(+∞) ((cos(ch(nx)))/((3+x^2 )^2 ))dx 1) calculate U_n interms of n 2) find lim_(n→+∞) n U_n and lim_(n→+∞) n^2 U_n 3)study the serie Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{cos}\left({ch}\left({nx}\right)\right)}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{U}_{{n}} \:\:\:\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 62826    Answers: 0   Comments: 4

Question Number 62821    Answers: 2   Comments: 1

Question Number 62815    Answers: 0   Comments: 2

developp at fourier serie f(x) =cos(tx) ,2π periodic even .

$${developp}\:{at}\:{fourier}\:{serie}\:{f}\left({x}\right)\:={cos}\left({tx}\right)\:\:,\mathrm{2}\pi\:{periodic}\:{even}\:. \\ $$

Question Number 62814    Answers: 0   Comments: 10

Question Number 62813    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt study first the convergence .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({t}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)} {dt} \\ $$$${study}\:{first}\:{the}\:{convergence}\:. \\ $$

Question Number 62812    Answers: 0   Comments: 1

let U_n =∫_0 ^(+∞) ((arctan(nt))/(1+n^2 t^2 ))dt with n natural≥1 1) calculate U_n 2) calculate lim_(n→+∞) n^2 U_n 3) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left({nt}\right)}{\mathrm{1}+{n}^{\mathrm{2}} {t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:{n}\:{natural}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 62811    Answers: 0   Comments: 2

1) find ∫ ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx 2)calculate ∫_5 ^(+∞) ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx

$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$

Question Number 62809    Answers: 0   Comments: 1

let f(x) = arctan(nx) with n integr natural 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie .

$${let}\:{f}\left({x}\right)\:=\:{arctan}\left({nx}\right)\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 62808    Answers: 0   Comments: 0

f(t) =∫_0 ^(+∞) (e^(−xt) /((x+t)^2 ))dx with t≥0 1) study the set of definition for f(t) 2)study the continuity of f 3)study the derivability of f 4) developp f at integr serie

$${f}\left({t}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{xt}} }{\left({x}+{t}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{set}\:{of}\:{definition}\:{for}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{derivability}\:{of}\:{f} \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 62806    Answers: 0   Comments: 1

find the value of ∫_(−∞) ^(+∞) ((x+1)/((x^4 +x^2 +1)^3 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 62805    Answers: 0   Comments: 1

calculate ∫_0 ^(+∞) ((3x^2 −2)/((x^2 +1)( x^2 −2i)^2 )) dx

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\:{x}^{\mathrm{2}} −\mathrm{2}{i}\right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 63060    Answers: 1   Comments: 0

Question Number 62801    Answers: 0   Comments: 0

Question Number 62800    Answers: 0   Comments: 2

Question Number 62798    Answers: 0   Comments: 0

Calculate tg(20°)+4sin(20°)+1

$$\boldsymbol{\mathrm{Calculate}} \\ $$$$\boldsymbol{\mathrm{tg}}\left(\mathrm{20}°\right)+\mathrm{4}\boldsymbol{\mathrm{sin}}\left(\mathrm{20}°\right)+\mathrm{1} \\ $$

Question Number 62790    Answers: 1   Comments: 1

Question Number 62782    Answers: 0   Comments: 0

Question Number 62781    Answers: 0   Comments: 0

∫ sec(2x) e^(2x) dx

$$\int\:\mathrm{sec}\left(\mathrm{2x}\right)\:\mathrm{e}^{\mathrm{2x}} \:\:\mathrm{dx} \\ $$

Question Number 62777    Answers: 0   Comments: 1

5^(3x−1) .4^(2x−2) =625

$$\mathrm{5}^{\mathrm{3}{x}−\mathrm{1}} .\mathrm{4}^{\mathrm{2}{x}−\mathrm{2}} =\mathrm{625} \\ $$

Question Number 62763    Answers: 0   Comments: 0

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