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Question Number 29157 Answers: 1 Comments: 1

find lim_(x→1) (((x−2)^(1/3) +(1−x+x^2 )^(1/3) )/(x^2 −1)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\:\frac{\left({x}−\mathrm{2}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:\:+\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{{x}^{\mathrm{2}} −\mathrm{1}}\:. \\ $$

Question Number 29156 Answers: 0 Comments: 2

find lim_(x→0^+ ) x[(1/x)] and lim_(x→0^+ ) x^2 [ (1/x)] . [α] is the greatest integr inferior or equal to α.

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:{x}\left[\frac{\mathrm{1}}{{x}}\right]\:\:{and}\:\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\:{x}^{\mathrm{2}} \:\left[\:\frac{\mathrm{1}}{{x}}\right]\:\:. \\ $$$$\left[\alpha\right]\:{is}\:{the}\:{greatest}\:{integr}\:{inferior}\:{or}\:{equal}\:{to}\:\alpha. \\ $$

Question Number 29123 Answers: 0 Comments: 0

Question Number 29118 Answers: 1 Comments: 6

Question Number 29116 Answers: 1 Comments: 0

Find the area of the region R bounded by the curve y = cosh(x), the line x = log_e (2) and the coordinate axis . Find also the volume obtained when R is rotated completely about the x − axis.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region}\:\boldsymbol{\mathrm{R}}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curve}\:\:\mathrm{y}\:=\:\mathrm{cosh}\left(\mathrm{x}\right),\:\:\mathrm{the}\:\mathrm{line}\:\:\mathrm{x}\:=\:\mathrm{log}_{\mathrm{e}} \left(\mathrm{2}\right) \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{coordinate}\:\mathrm{axis}\:.\:\:\mathrm{Find}\:\mathrm{also}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{obtained}\:\mathrm{when}\:\boldsymbol{\mathrm{R}}\:\mathrm{is}\:\mathrm{rotated}\: \\ $$$$\mathrm{completely}\:\mathrm{about}\:\mathrm{the}\:\:\mathrm{x}\:−\:\mathrm{axis}. \\ $$

Question Number 29111 Answers: 1 Comments: 1

cos^3 x.sin^2 x=1/16(2cos x−cos 3x−cos 5x)

$$\mathrm{cos}^{\mathrm{3}} {x}.\mathrm{sin}\:^{\mathrm{2}} {x}=\mathrm{1}/\mathrm{16}\left(\mathrm{2cos}\:{x}−\mathrm{cos}\:\mathrm{3}{x}−\mathrm{cos}\:\mathrm{5}{x}\right) \\ $$

Question Number 29107 Answers: 1 Comments: 1

Question Number 29105 Answers: 0 Comments: 2

Show that: ∫_(−1) ^( 1) (dx/(5 cosh(x) + 13 sinh(x))) = (1/2) log_e (((15e − 10)/(3e + 2)))

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\int_{−\mathrm{1}} ^{\:\:\:\mathrm{1}} \:\:\:\:\:\:\:\frac{\mathrm{dx}}{\mathrm{5}\:\mathrm{cosh}\left(\mathrm{x}\right)\:+\:\mathrm{13}\:\mathrm{sinh}\left(\mathrm{x}\right)}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{log}_{\mathrm{e}} \left(\frac{\mathrm{15e}\:−\:\mathrm{10}}{\mathrm{3e}\:+\:\mathrm{2}}\right)\: \\ $$

Question Number 29080 Answers: 2 Comments: 0

let give a>0 study the convergence of Σ_(n=1) ^∞ a^H_n with H_n = Σ_(k=1) ^n (1/k).

$${let}\:{give}\:{a}>\mathrm{0}\:{study}\:{the}\:{convergence}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:{a}^{{H}_{{n}} } \:\: \\ $$$${with}\:{H}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}}. \\ $$

Question Number 29079 Answers: 0 Comments: 0

let give w(x)= ∫_0 ^1 ((arcsin(x(1+t^2 )))/(1+t^2 ))dt find w(x).

$${let}\:{give}\:{w}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{arcsin}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{w}\left({x}\right). \\ $$

Question Number 29078 Answers: 0 Comments: 2

let give h(x)= ∫_0 ^1 ((arctan(xt))/(1+t^2 )) find h(x) .

$${let}\:{give}\:\:{h}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{find}\:{h}\left({x}\right)\:. \\ $$

Question Number 29077 Answers: 0 Comments: 1

let give g(x)=∫_0 ^∞ ((arctan(x(1+t^2 )))/(1+t^2 ))dt find a simple form of g^′ (x) without integral.

$${let}\:{give}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:\:{g}^{'} \left({x}\right)\:{without}\:{integral}. \\ $$

Question Number 29076 Answers: 0 Comments: 1

let give f(x)= ∫_0 ^1 ((arctan(x(1+t^2 )))/(1+t^2 ))dt find asimple form of f(x) without integral.

$${let}\:{give}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{asimple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:{without}\:{integral}. \\ $$

Question Number 29063 Answers: 0 Comments: 0

Question Number 29049 Answers: 1 Comments: 0

Prove that e^(iπ) +1=0

$${Prove}\:{that} \\ $$$$\:\:\:\:\:{e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$

Question Number 29048 Answers: 1 Comments: 19

Question Number 29043 Answers: 0 Comments: 1

∫tan^− (1−sinx/1+sinx) dx

$$\int\mathrm{tan}^{−} \left(\mathrm{1}−\mathrm{sinx}/\mathrm{1}+\mathrm{sinx}\right)\:\mathrm{dx} \\ $$

Question Number 29038 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((cos(at))/(1+t^4 ))dt.

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({at}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 29037 Answers: 0 Comments: 0

let give the sequence (y_n ) /y_0 (x)=1 and y_n (x)= 1+ ∫_0 ^x (y_(n−1) (t))^2 dt , let suppose x∈[0,1] prove that (y_n ) is increasing majored by (1/(1−x)) if y=lim_(n→+∞) y_n prove that y is solution of differencial equation y^, =y^2 and y(o)=1.

$${let}\:{give}\:{the}\:{sequence}\:\:\left({y}_{{n}} \right)\:/{y}_{\mathrm{0}} \left({x}\right)=\mathrm{1}\:\:{and} \\ $$$${y}_{{n}} \left({x}\right)=\:\mathrm{1}+\:\int_{\mathrm{0}} ^{{x}} \left({y}_{{n}−\mathrm{1}} \left({t}\right)\right)^{\mathrm{2}} {dt}\:,\:{let}\:{suppose}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{prove} \\ $$$${that}\:\left({y}_{{n}} \right)\:{is}\:{increasing}\:{majored}\:{by}\:\frac{\mathrm{1}}{\mathrm{1}−{x}}\:{if}\:{y}={lim}_{{n}\rightarrow+\infty} {y}_{{n}} \\ $$$${prove}\:{that}\:{y}\:{is}\:{solution}\:{of}\:{differencial}\:{equation} \\ $$$${y}^{,} ={y}^{\mathrm{2}} \:{and}\:{y}\left({o}\right)=\mathrm{1}. \\ $$

Question Number 29036 Answers: 0 Comments: 0

p=2m+1 is a prime number prove that 1) (p−1)!≡ −1[p] 2) (m!)^2 ≡ (−1)^(m+1) [p]

$${p}=\mathrm{2}{m}+\mathrm{1}\:{is}\:{a}\:{prime}\:{number}\:{prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\left({p}−\mathrm{1}\right)!\equiv\:−\mathrm{1}\left[{p}\right] \\ $$$$\left.\mathrm{2}\right)\:\left({m}!\right)^{\mathrm{2}} \equiv\:\left(−\mathrm{1}\right)^{{m}+\mathrm{1}} \:\left[{p}\right] \\ $$

Question Number 29035 Answers: 0 Comments: 0

let give a prime number p>2 and a /D(a,p)=1 and suppose that the equation x^2 ≡ a[p]have a solution1) 1) prove that a^((p−1)/2) ≡ 1 [p] 2)prove that x^2 ≡ −1[p] ⇔ p≡ 1 [4]

$${let}\:{give}\:{a}\:{prime}\:{number}\:{p}>\mathrm{2}\:\:{and}\:{a}\:/{D}\left({a},{p}\right)=\mathrm{1}\:{and}\: \\ $$$$\left.{suppose}\:{that}\:{the}\:{equation}\:{x}^{\mathrm{2}} \equiv\:{a}\left[{p}\right]{have}\:{a}\:{solution}\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:{a}^{\frac{{p}−\mathrm{1}}{\mathrm{2}}} \:\:\:\equiv\:\mathrm{1}\:\left[{p}\right] \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:{x}^{\mathrm{2}} \equiv\:−\mathrm{1}\left[{p}\right]\:\Leftrightarrow\:\:\:{p}\equiv\:\mathrm{1}\:\left[\mathrm{4}\right] \\ $$

Question Number 29032 Answers: 0 Comments: 0

let give A = (((0 1 0)),((0 0 1)) ) (1 0 0 1) find A^3 2) find e^(tA) .

$$ \\ $$$${let}\:{give}\:{A}\:=\:\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{0}\right. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}^{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{tA}} \:\:. \\ $$

Question Number 29031 Answers: 0 Comments: 0

find all function f ∈C^1 (R^2 ,R) wich verify (∂f/∂x) −(∂f/∂y)=0 ∀(x,y)∈R^2 .

$${find}\:{all}\:{function}\:{f}\:\in{C}^{\mathrm{1}} \left({R}^{\mathrm{2}} ,{R}\right)\:{wich}\:{verify} \\ $$$$\frac{\partial{f}}{\partial{x}}\:−\frac{\partial{f}}{\partial{y}}=\mathrm{0}\:\:\:\forall\left({x},{y}\right)\in{R}^{\mathrm{2}} . \\ $$

Question Number 29030 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (t^(3n) /((3n)!)) .

$${find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{t}^{\mathrm{3}{n}} }{\left(\mathrm{3}{n}\right)!}\:. \\ $$

Question Number 29029 Answers: 0 Comments: 0

let give A= (((1 −1)),((4 −3)) ) calculate A^n and e^A .

$${let}\:{give}\:{A}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{4}\:\:\:\:\:\:\:−\mathrm{3}}\end{pmatrix}\:\:{calculate}\:{A}^{{n}} \:{and}\:{e}^{{A}} . \\ $$

Question Number 29028 Answers: 0 Comments: 0

for t>0 and f(t)= (4πt)^(−(n/2)) e^(−(x^2 /(4t))) prove that ∫_R f_t (x)dx=1 ∀t>0.

$${for}\:{t}>\mathrm{0}\:\:{and}\:{f}\left({t}\right)=\:\left(\mathrm{4}\pi{t}\right)^{−\frac{{n}}{\mathrm{2}}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{4}{t}}} \:\:\:{prove}\:{that} \\ $$$$\int_{{R}} {f}_{{t}} \left({x}\right){dx}=\mathrm{1}\:\:\:\forall{t}>\mathrm{0}. \\ $$

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