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Question Number 74352 Answers: 1 Comments: 0

let U_n =Σ_(k=0) ^n (1/(k^2 +k+1)) find a equivalent of U_n (n→+∞)

$${let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{k}^{\mathrm{2}} +{k}+\mathrm{1}}\:\:{find}\:{a}\:{equivalent}\:{of}\:{U}_{{n}} \:\:\:\left({n}\rightarrow+\infty\right) \\ $$$$ \\ $$

Question Number 74351 Answers: 0 Comments: 1

find the value of Σ_(n=1) ^(+∞) (((−1)^n )/((4n^2 −1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 74350 Answers: 0 Comments: 1

findf(a)= ∫_(−∞) ^(+∞) ((arctan(cosx))/(x^2 +a^2 ))dx witha>0

$${findf}\left({a}\right)=\:\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left({cosx}\right)}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}\:{witha}>\mathrm{0} \\ $$

Question Number 74349 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((cos(2πx))/((x^2 +3)^2 ))dx

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

Question Number 74348 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((arctan(sin(x^2 )))/(x^2 +1))dx

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

Question Number 74347 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(cos(πx^2 )))/(x^2 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left({cos}\left(\pi{x}^{\mathrm{2}} \right)\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 74346 Answers: 1 Comments: 0

find ∫ ((x+(√(x+1)))/(2(√(x−1))+3))dx

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

Question Number 74345 Answers: 0 Comments: 2

1) calculate f(x)=∫_(x+1) ^(x^2 +1) e^(−xt) arctan(t)dt 2) find lim_(x→0) f(x)

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right)=\int_{{x}+\mathrm{1}} ^{{x}^{\mathrm{2}} +\mathrm{1}} \:\:\:{e}^{−{xt}} {arctan}\left({t}\right){dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:{f}\left({x}\right) \\ $$

Question Number 74344 Answers: 1 Comments: 1

calculate ∫ ((x^2 −x+3)/(x^3 (x+2)^2 ))dx

$${calculate}\:\int\:\:\:\:\:\frac{{x}^{\mathrm{2}} −{x}+\mathrm{3}}{{x}^{\mathrm{3}} \left({x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 74343 Answers: 0 Comments: 1

calculatef(α)= ∫_0 ^∞ ((arctan(αx^2 ))/(x^2 +9))dx with α real.

$${calculatef}\left(\alpha\right)=\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\alpha{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx}\:\:\:{with}\:\alpha\:{real}. \\ $$

Question Number 74342 Answers: 1 Comments: 2

1) calculate U_n =∫_0 ^∞ e^(−nx) [x]dx 2) find lim_(n→+∞) n U_n 3) determine nsture of the serie Σ U_n

$$\left.\mathrm{1}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}} \left[{x}\right]{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{lim}_{{n}\rightarrow+\infty} \:\:{n}\:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{nsture}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 74335 Answers: 1 Comments: 0

5(1/2)×(6/7)=? The end result must in the mixed fraction.

$$\mathrm{5}\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{6}}{\mathrm{7}}=?\: \\ $$$${The}\:{end}\:{result}\:{must}\:{in}\:{the} \\ $$$$\boldsymbol{{mixed}}\:\boldsymbol{{fraction}}. \\ $$

Question Number 74334 Answers: 0 Comments: 1

∫te^t cos e^t .e^t dt

$$\int{te}^{{t}} \mathrm{cos}\:{e}^{{t}} .{e}^{{t}} {dt} \\ $$

Question Number 74371 Answers: 2 Comments: 1

Question Number 74329 Answers: 1 Comments: 1

Solve : ax+by=r bx−ay=s

$${Solve}\:: \\ $$$${ax}+{by}={r} \\ $$$${bx}−{ay}={s} \\ $$

Question Number 74328 Answers: 1 Comments: 0

Question Number 74323 Answers: 0 Comments: 3

Question Number 74322 Answers: 1 Comments: 0

Let k = (((xy + yz + zx)(x + y + z))/((x + y)(y + z)(z + x))) Find the minimum and maximum value of k .

$${Let}\:\: \\ $$$${k}\:\:=\:\:\frac{\left({xy}\:+\:{yz}\:+\:{zx}\right)\left({x}\:+\:{y}\:+\:{z}\right)}{\left({x}\:+\:{y}\right)\left({y}\:+\:{z}\right)\left({z}\:+\:{x}\right)} \\ $$$${Find}\:\:{the}\:\:{minimum}\:\:{and}\:\:{maximum}\:\:{value}\:\:{of}\:\:\:{k}\:. \\ $$

Question Number 74320 Answers: 0 Comments: 0

∫_0 ^x x e^x (cos e^x )e^x dx

$$\int_{\mathrm{0}} ^{{x}} {x}\:{e}^{{x}} \left(\mathrm{cos}\:\:{e}^{{x}} \right){e}^{{x}} {dx} \\ $$

Question Number 74359 Answers: 0 Comments: 1

Question Number 74358 Answers: 0 Comments: 6

Question Number 74308 Answers: 1 Comments: 0

Question Number 74301 Answers: 1 Comments: 0

Prove that S={(x,y,z)∈R^3 \ x^2 +y^2 =z^2 } is a surface and find out if possible the tangent plan in O(0,0,0).

$${Prove}\:{that}\:\:{S}=\left\{\left({x},{y},{z}\right)\in\mathbb{R}^{\mathrm{3}} \backslash\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={z}^{\mathrm{2}} \:\right\}\:{is}\:{a}\:{surface}\: \\ $$$${and}\:{find}\:{out}\:{if}\:{possible}\:{the}\:{tangent}\:{plan}\:{in}\:{O}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right). \\ $$

Question Number 74300 Answers: 1 Comments: 0

Let consider γ :I→R^2 a parametric curve 1)Prove that if a<b and γ(a)≠γ(b) then there exist t_0 ∈]a,b[ such as γ′(t_0 ) is colinear to γ(b)−γ(a) 2)Show that if γ is regular and the function f :I→R t→f(t)=∣∣γ(t)−O(0,0) ∣∣ is maximal in t_0 ∈I Then ∣K_γ (t_0 )∣≥(1/(f(t_0 )))

$${Let}\:{consider}\:\:\gamma\:\::{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\: \\ $$$$\left.\mathrm{1}\left.\right){Prove}\:{that}\:{if}\:\:{a}<{b}\:\:{and}\:\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{then}\:{there}\:{exist}\:\:{t}_{\mathrm{0}} \in\right]{a},{b}\left[\:\:\right. \\ $$$${such}\:{as}\:\:\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right){Show}\:{that}\:{if}\:\:\gamma\:{is}\:{regular}\:{and}\:{the}\:\:{function}\:{f}\::{I}\rightarrow\mathbb{R}\:\:\:\:{t}\rightarrow{f}\left({t}\right)=\mid\mid\gamma\left({t}\right)−{O}\left(\mathrm{0},\mathrm{0}\right)\:\mid\mid\:\:{is}\:{maximal}\:{in}\:{t}_{\mathrm{0}} \in{I} \\ $$$${Then}\:\:\mid{K}_{\gamma} \left({t}_{\mathrm{0}} \right)\mid\geqslant\frac{\mathrm{1}}{{f}\left({t}_{\mathrm{0}} \right)} \\ $$

Question Number 74293 Answers: 0 Comments: 2

Question Number 74284 Answers: 2 Comments: 4

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