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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

Question Number 62762    Answers: 0   Comments: 0

Question Number 62761    Answers: 0   Comments: 3

if cosθ=((cosA.cosB)/(1−cosA.cosB)) prove that Tan(θ/2)=Tan(A/2).Cot(B/2)

$$\mathrm{if}\:\mathrm{cos}\theta=\frac{\mathrm{cosA}.\mathrm{cosB}}{\mathrm{1}−\mathrm{cosA}.\mathrm{cosB}}\:\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{Tan}\frac{\theta}{\mathrm{2}}=\mathrm{Tan}\frac{\mathrm{A}}{\mathrm{2}}.\mathrm{Cot}\frac{\mathrm{B}}{\mathrm{2}} \\ $$

Question Number 62759    Answers: 1   Comments: 0

y = x^y (dy/dx) = ?

$${y}\:\:=\:\:{x}^{{y}} \\ $$$$\frac{{dy}}{{dx}}\:\:=\:\:? \\ $$

Question Number 62754    Answers: 1   Comments: 2

1)∫(dx/(1−sin(x))) R solve in(2) (4−x)^4 +x^4 =82

$$\left.\mathrm{1}\right)\int\frac{{dx}}{\mathrm{1}−{sin}\left({x}\right)} \\ $$$$ \\ $$$${R}\:{solve}\:{in}\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\left(\mathrm{4}−{x}\right)^{\mathrm{4}} +{x}^{\mathrm{4}} =\mathrm{82} \\ $$

Question Number 62753    Answers: 1   Comments: 0

The normal at the point P(4cos θ,3sin θ) on the ellipse (x^2 /(16)) +(y^2 /9)=1 meets the x−axis and y−axis at A and B respectively show that locus of the mid−point of AB is an ellipse with the same eccentricity as given ellipse.

$${The}\:{normal}\:{at}\:{the}\:{point} \\ $$$${P}\left(\mathrm{4cos}\:\theta,\mathrm{3sin}\:\theta\right)\:{on}\:{the} \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{\mathrm{16}}\:+\frac{{y}^{\mathrm{2}} }{\mathrm{9}}=\mathrm{1}\:{meets} \\ $$$${the}\:{x}−{axis}\:{and}\:{y}−{axis} \\ $$$${at}\:{A}\:{and}\:{B}\:{respectively} \\ $$$${show}\:{that}\:{locus}\:{of}\:{the} \\ $$$${mid}−{point}\:{of}\:{AB}\:{is}\:{an} \\ $$$${ellipse}\:{with}\:{the}\:{same} \\ $$$${eccentricity}\:{as}\:{given} \\ $$$${ellipse}. \\ $$

Question Number 62747    Answers: 1   Comments: 0

Are f, g: R→R defined by f(x)= { ((0, x ∈ R\Q)),((x, x ∈Q)) :} g(x)= { ((1, x=0)),((0, x≠0)) :} show that lim_(x→0) f(x)=0 and lim_(y→0) g(y)=0 however lim_(x→0) g(f(x)) does not exist.

$${Are}\:\boldsymbol{{f}},\:\boldsymbol{{g}}:\:\mathbb{R}\rightarrow\mathbb{R}\:{defined}\:{by} \\ $$$${f}\left({x}\right)=\begin{cases}{\mathrm{0},\:\:\:{x}\:\in\:\mathbb{R}\backslash\mathbb{Q}}\\{{x},\:\:\:{x}\:\in\mathbb{Q}}\end{cases} \\ $$$${g}\left({x}\right)=\begin{cases}{\mathrm{1},\:\:\:{x}=\mathrm{0}}\\{\mathrm{0},\:\:\:\:{x}\neq\mathrm{0}}\end{cases} \\ $$$${show}\:{that}\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\boldsymbol{\mathrm{lim}}}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{0}\:{and}\:\underset{\boldsymbol{\mathrm{y}}\rightarrow\mathrm{0}} {\boldsymbol{\mathrm{lim}}}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{y}}\right)=\mathrm{0} \\ $$$${however}\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\boldsymbol{\mathrm{lim}}}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right)\:{does}\:{not}\:{exist}. \\ $$

Question Number 62735    Answers: 1   Comments: 1

Question Number 62732    Answers: 1   Comments: 0

calculate ∫_0 ^(2π) ((cos(2x))/(2cosx −sin(x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}{cosx}\:−{sin}\left({x}\right)}{dx}\: \\ $$

Question Number 62731    Answers: 0   Comments: 1

find ∫ (√((x−1)/(x^2 +3)))dx

$${find}\:\int\:\sqrt{\frac{{x}−\mathrm{1}}{{x}^{\mathrm{2}} \:+\mathrm{3}}}{dx}\: \\ $$

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