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Question Number 219579 Answers: 0 Comments: 0

∫_0 ^( ∞) sin(t)J_ν (kt) dt ∫_0 ^( ∞) t∙J_ν (at)J_ν (kt) dt

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\mathrm{sin}\left({t}\right){J}_{\nu} \left({kt}\right)\:\mathrm{d}{t} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:{t}\centerdot{J}_{\nu} \left({at}\right){J}_{\nu} \left({kt}\right)\:\mathrm{d}{t} \\ $$

Question Number 219577 Answers: 0 Comments: 1

prove ∫_0 ^( 1) ∫_0 ^( 1) ...∫_0 ^( 1) _(n times) x_1 ^α x_2 ^α ....x_n ^α ln(x_1 )ln(x_2 )....ln(x_n )dx_1 dx_2 ..dx_n =^(Equal) (((−)^n )/((α+1)^(2n) ))

$$\mathrm{prove} \\ $$$$\underset{{n}\:\mathrm{times}} {\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} ...\int_{\mathrm{0}} ^{\:\mathrm{1}} }\:\:{x}_{\mathrm{1}} ^{\alpha} {x}_{\mathrm{2}} ^{\alpha} ....{x}_{{n}} ^{\alpha} \mathrm{ln}\left({x}_{\mathrm{1}} \right)\mathrm{ln}\left({x}_{\mathrm{2}} \right)....\mathrm{ln}\left({x}_{{n}} \right)\mathrm{d}{x}_{\mathrm{1}} \mathrm{d}{x}_{\mathrm{2}} ..\mathrm{d}{x}_{{n}} \\ $$$$\overset{\mathrm{Equal}} {=}\:\:\:\frac{\left(−\right)^{{n}} }{\left(\alpha+\mathrm{1}\right)^{\mathrm{2}{n}} } \\ $$

Question Number 219574 Answers: 0 Comments: 0

∫_0 ^( ∞) ((sin(z))/(z(z^2 +4))) dz=(1/4)∫_0 ^( ∞) (((sin(z))/z)−((sin(z))/(2z+4i))−((sin(z))/(2z−4i))) dz and next....??? 2πiΣ_(j=1) ^M Res_(h=a_j ) {f(h)}....

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({z}\right)}{{z}\left({z}^{\mathrm{2}} +\mathrm{4}\right)}\:\mathrm{d}{z}=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\:\infty} \:\:\left(\frac{\mathrm{sin}\left({z}\right)}{{z}}−\frac{\mathrm{sin}\left({z}\right)}{\mathrm{2}{z}+\mathrm{4}\boldsymbol{{i}}}−\frac{\mathrm{sin}\left({z}\right)}{\mathrm{2}{z}−\mathrm{4}\boldsymbol{{i}}}\right)\:\mathrm{d}{z} \\ $$$$\mathrm{and}\:\mathrm{next}....??? \\ $$$$\mathrm{2}\pi\boldsymbol{{i}}\underset{{j}=\mathrm{1}} {\overset{{M}} {\sum}}\:\:\mathrm{Res}_{{h}={a}_{{j}} } \left\{{f}\left({h}\right)\right\}.... \\ $$

Question Number 219571 Answers: 1 Comments: 0

pls Help.....! prove ∫∫_( S) g^→ ∙dS^→ =0 ⇆ div g^→ =0

$$\mathrm{pls}\:\mathrm{Help}.....! \\ $$$$\mathrm{prove} \\ $$$$\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{g}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=\mathrm{0}\:\leftrightarrows\:\mathrm{div}\:\overset{\rightarrow} {\boldsymbol{\mathrm{g}}}=\mathrm{0} \\ $$

Question Number 219570 Answers: 2 Comments: 0

8x^2 +9y^2 =36(x+y), x,y∈R, find maximum (x+y)

$$\:\mathrm{8}{x}^{\mathrm{2}} +\mathrm{9}{y}^{\mathrm{2}} =\mathrm{36}\left({x}+{y}\right),\: \\ $$$$\:\:{x},{y}\in{R},\:{find}\:{maximum}\:\left({x}+{y}\right) \\ $$$$\: \\ $$

Question Number 219564 Answers: 0 Comments: 0

solve for real numbers: { ((((xy(x^3 −y^3 )+yz(y^3 −z^3 )+zx(z^3 −x^3 ))/(xy(x−y)+yz(y−z)+zx(z−x))) = 55)),((x^3 + y^3 + z^3 = 99)),((((xy(x^3 −y^3 )+yz(y^3 −z^3 )+zx(z^3 −x^3 ))/(xy(x^2 −y^2 )+yz(y^2 −z^2 )+zx(z^2 −y^2 ))) = ((55)/9))) :}

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\frac{\mathrm{xy}\left(\mathrm{x}^{\mathrm{3}} −\mathrm{y}^{\mathrm{3}} \right)+\mathrm{yz}\left(\mathrm{y}^{\mathrm{3}} −\mathrm{z}^{\mathrm{3}} \right)+\mathrm{zx}\left(\mathrm{z}^{\mathrm{3}} −\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{xy}\left(\mathrm{x}−\mathrm{y}\right)+\mathrm{yz}\left(\mathrm{y}−\mathrm{z}\right)+\mathrm{zx}\left(\mathrm{z}−\mathrm{x}\right)}\:=\:\mathrm{55}}\\{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:=\:\mathrm{99}}\\{\frac{\mathrm{xy}\left(\mathrm{x}^{\mathrm{3}} −\mathrm{y}^{\mathrm{3}} \right)+\mathrm{yz}\left(\mathrm{y}^{\mathrm{3}} −\mathrm{z}^{\mathrm{3}} \right)+\mathrm{zx}\left(\mathrm{z}^{\mathrm{3}} −\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{xy}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)+\mathrm{yz}\left(\mathrm{y}^{\mathrm{2}} −\mathrm{z}^{\mathrm{2}} \right)+\mathrm{zx}\left(\mathrm{z}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)}\:=\:\frac{\mathrm{55}}{\mathrm{9}}}\end{cases} \\ $$

Question Number 219563 Answers: 2 Comments: 0

find all n ∈ N^∗ such that ∫_0 ^( 1) (sinx)^(2n−2) ∙ (cosx)^(2n) dx ≥ (1/4^(1011) )

$$\mathrm{find}\:\mathrm{all}\:\:\:\mathrm{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\mathrm{sinx}\right)^{\mathrm{2n}−\mathrm{2}} \:\centerdot\:\left(\mathrm{cosx}\right)^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{1011}} } \\ $$

Question Number 219562 Answers: 1 Comments: 0

prove that exists X ∈ M_(2,3) (R) Y ∈ M_(3,2) (R) such that X∙Y = ((1,1),(1,1) ) Y∙X = ((2,6,6),(3,9,9),((-3),(-9),(-9)) )

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{exists}\:\:\:\mathrm{X}\:\in\:\mathrm{M}_{\mathrm{2},\mathrm{3}} \:\left(\mathbb{R}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\:\in\:\mathrm{M}_{\mathrm{3},\mathrm{2}} \:\left(\mathbb{R}\right) \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\:\mathrm{X}\centerdot\mathrm{Y}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}}\end{pmatrix}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\centerdot\mathrm{X}\:=\:\begin{pmatrix}{\mathrm{2}}&{\mathrm{6}}&{\mathrm{6}}\\{\mathrm{3}}&{\mathrm{9}}&{\mathrm{9}}\\{-\mathrm{3}}&{-\mathrm{9}}&{-\mathrm{9}}\end{pmatrix}\: \\ $$

Question Number 219561 Answers: 0 Comments: 3

let be the sequence (x_n )n ≥ 1 defined by x_1 = 1 x_(n+2) = 3x_(n+1) − x_n ∀n ∈ N find L =lim_(n→∞) ((Σ_(k=0) ^0 (x_(2k+1) /(x_k + x_(k+1) ))))^(1/n) = ?

$$\mathrm{let}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sequence}\:\:\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\mathrm{n}\:\geqslant\:\mathrm{1} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\:\:\mathrm{x}_{\mathrm{1}} =\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}+\mathrm{2}} \:=\:\mathrm{3x}_{\boldsymbol{\mathrm{n}}+\mathrm{1}} −\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \\ $$$$\forall\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{find}\:\:\:\boldsymbol{\mathrm{L}}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\mathrm{0}} {\sum}}\:\:\frac{\mathrm{x}_{\mathrm{2}\boldsymbol{\mathrm{k}}+\mathrm{1}} }{\mathrm{x}_{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{x}_{\boldsymbol{\mathrm{k}}+\mathrm{1}} }}\:=\:? \\ $$

Question Number 219560 Answers: 0 Comments: 0

Suppose that an urn contains 100,000 marbles and 120 are red . If a random sample of 1000 is drawn, what are the probabilities that 0,1,2,3 and 4 respectively will be red. What is the mean and variance?

Question Number 219556 Answers: 0 Comments: 0