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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.

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

Question Number 29027    Answers: 0   Comments: 0

find ∫∫_D e^(−y) sin(2xy)dxdy with D=[0,1]×[0,+∞[ then find the value of ∫_0 ^∞ ((sin^2 t)/t) e^(−t) dt .

$${find}\:\int\int_{{D}} \:{e}^{−{y}} {sin}\left(\mathrm{2}{xy}\right){dxdy}\:{with}\:{D}=\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},+\infty\left[\right.\right. \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{\mathrm{2}} {t}}{{t}}\:{e}^{−{t}} {dt}\:\:. \\ $$

Question Number 29093    Answers: 1   Comments: 1

Question Number 29016    Answers: 1   Comments: 1

Question Number 29018    Answers: 0   Comments: 0

∫ (√(Σ_(n = 0) ^∞ [(−1)^n tan^(2n) (2x)])) dx

$$\int\:\sqrt{\underset{{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\left[\left(−\mathrm{1}\right)^{{n}} \:\mathrm{tan}^{\mathrm{2}{n}} \:\left(\mathrm{2}{x}\right)\right]}\:{dx} \\ $$

Question Number 29014    Answers: 0   Comments: 3

Prove that A∪A^c =A

$${Prove}\:\:{that}\:{A}\cup{A}^{{c}} ={A} \\ $$

Question Number 29007    Answers: 0   Comments: 1

Question Number 29003    Answers: 1   Comments: 1

find ∫_0 ^∞ (dx/(1+x^3 )) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} }\:. \\ $$

Question Number 29002    Answers: 0   Comments: 0

let give 0<p<1 calculate K(p)= ∫_(−∞) ^(+∞) (e^(pt) /(1+e^t ))dt.

$${let}\:{give}\:\mathrm{0}<{p}<\mathrm{1}\:{calculate}\:\:{K}\left({p}\right)=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{pt}} }{\mathrm{1}+{e}^{{t}} }{dt}. \\ $$

Question Number 29001    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ ((cos(ξt))/(1+t^4 ))dt.

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\xi{t}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 29000    Answers: 0   Comments: 1

prove thst ∫_R (e^(iξx) /(1+x^2 ))dx= π e^(−∣ξ∣) .

$${prove}\:{thst}\:\:\:\:\int_{\mathbb{R}} \:\:\:\:\frac{{e}^{{i}\xi{x}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\:\pi\:{e}^{−\mid\xi\mid} \:\:. \\ $$

Question Number 28999    Answers: 0   Comments: 1

prove that ∫_0 ^∞ (e^(−t) /(√t))dt= e^(i(π/4)) ∫_0 ^∞ (e^(−ix) /(√x))dx.

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} }{\sqrt{{t}}}{dt}=\:{e}^{{i}\frac{\pi}{\mathrm{4}}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{ix}} }{\sqrt{{x}}}{dx}. \\ $$

Question Number 28998    Answers: 0   Comments: 0

find ∫_γ (e^z /(z(z+1)))dz with γ={z∈C/ ∣z−1∣=2}

$${find}\:\int_{\gamma} \:\:\:\:\frac{{e}^{{z}} }{{z}\left({z}+\mathrm{1}\right)}{dz}\:{with}\:\gamma=\left\{{z}\in{C}/\:\mid{z}−\mathrm{1}\mid=\mathrm{2}\right\} \\ $$

Question Number 28997    Answers: 0   Comments: 1

find ∫_(−∞) ^(+∞) (dx/((1+x^2 )( 2+e^(ix) ))) .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{2}+{e}^{{ix}} \right)}\:. \\ $$

Question Number 28996    Answers: 0   Comments: 0

find ∫_(−∞) ^(+∞) (x^2 /((x^2 +1)^2 (x^2 +2x+2)))dx.

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

Question Number 28995    Answers: 0   Comments: 0

find ∫_0 ^(2π) ((cos(2t))/(3−cost)) dt.

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{2}{t}\right)}{\mathrm{3}−{cost}}\:{dt}. \\ $$

Question Number 28994    Answers: 0   Comments: 0

find A_n = ∫_(−∞) ^(+∞) (dx/((1+x^2 )^n )) with n from N and n≥1.

$${find}\:\:{A}_{{n}} =\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:{with}\:{n}\:{from}\:{N}\:{and}\:{n}\geqslant\mathrm{1}. \\ $$

Question Number 28993    Answers: 1   Comments: 0

L means laplacr trsnsform find L (sin(at)) and L(cos(at)).

$${L}\:{means}\:{laplacr}\:{trsnsform}\:{find}\:{L}\:\left({sin}\left({at}\right)\right) \\ $$$${and}\:{L}\left({cos}\left({at}\right)\right). \\ $$

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