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Question Number 119762 Answers: 3 Comments: 0

calculate ∫_0 ^∞ ((x^4 dx)/((2x+1)^5 (3x+1)^8 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{x}^{\mathrm{4}} \mathrm{dx}}{\left(\mathrm{2x}+\mathrm{1}\right)^{\mathrm{5}} \left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{8}} } \\ $$

Question Number 119757 Answers: 0 Comments: 0

For any integer n, let I_n be the interval (n, n+1). Define R={(x, y)∈R∣both x, y ∈ I_n for some n∈Z} Then R is (A) reflexive on R (B) symmetric (C) transitive (D) an equivalence relation

$$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{n},\:\mathrm{let}\:{I}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{interval}\:\left({n},\:{n}+\mathrm{1}\right). \\ $$$$\mathrm{Define} \\ $$$$\:\:\:\:\:\:\:\mathcal{R}=\left\{\left(\mathrm{x},\:\mathrm{y}\right)\in\mathbb{R}\mid\mathrm{both}\:\mathrm{x},\:\mathrm{y}\:\in\:{I}_{{n}} \:\mathrm{for}\:\mathrm{some}\:{n}\in\mathbb{Z}\right\} \\ $$$$\mathrm{Then}\:\mathcal{R}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{reflexive}\:\mathrm{on}\:\mathbb{R} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{symmetric} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{transitive} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{an}\:\mathrm{equivalence}\:\mathrm{relation} \\ $$

Question Number 119755 Answers: 0 Comments: 2

Question Number 119754 Answers: 2 Comments: 3

find Σ_(n=1) ^∞ (u_n /(n!)) if u_n =u_(n+1) +u_(n−1)

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{u}_{{n}} }{{n}!}\:{if}\:\:{u}_{{n}} \:={u}_{{n}+\mathrm{1}} +{u}_{{n}−\mathrm{1}} \\ $$

Question Number 119752 Answers: 1 Comments: 0

find I_λ =∫_0 ^∞ ((ch(1+λcosx))/((x^2 +1)^2 ))dx (λ real >0)

$${find}\:{I}_{\lambda} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ch}\left(\mathrm{1}+\lambda{cosx}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\left(\lambda\:{real}\:>\mathrm{0}\right) \\ $$

Question Number 119750 Answers: 1 Comments: 0

Π_(k=1) ^(1019) [((2k)/(2k−1))]=?

$$\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{1019}} {\prod}}\left[\frac{\mathrm{2k}}{\mathrm{2k}−\mathrm{1}}\right]=? \\ $$

Question Number 119747 Answers: 1 Comments: 0

Suppose that 7 blue balls , 8 red balls and 9 green balls should be put into three boxes labeled 1,2 and 3, so that any box contains at least one balls of each colour. How many ways can this arrangement be done?

$${Suppose}\:{that}\:\mathrm{7}\:{blue}\:{balls}\:,\:\mathrm{8}\:{red}\:{balls}\:{and}\:\mathrm{9}\:{green} \\ $$$${balls}\:{should}\:{be}\:{put}\:{into}\:{three}\:{boxes}\:{labeled} \\ $$$$\mathrm{1},\mathrm{2}\:{and}\:\mathrm{3},\:{so}\:{that}\:{any}\:{box}\:{contains}\:{at}\:{least} \\ $$$${one}\:{balls}\:{of}\:{each}\:{colour}.\:{How}\:{many}\:{ways} \\ $$$${can}\:{this}\:{arrangement}\:{be}\:{done}? \\ $$

Question Number 119814 Answers: 3 Comments: 0

If a,b,c and d is real numbers satisfy (a/b)=(2/3), (c/d)=(4/5), (d/b)=(6/7) then (a/c) =?

$${If}\:{a},{b},{c}\:{and}\:{d}\:{is}\:{real}\:{numbers} \\ $$$${satisfy}\:\frac{{a}}{{b}}=\frac{\mathrm{2}}{\mathrm{3}},\:\frac{{c}}{{d}}=\frac{\mathrm{4}}{\mathrm{5}},\:\frac{{d}}{{b}}=\frac{\mathrm{6}}{\mathrm{7}} \\ $$$${then}\:\frac{{a}}{{c}}\:=? \\ $$

Question Number 119733 Answers: 1 Comments: 5

Question Number 119815 Answers: 1 Comments: 0

Given f(x+y)=4f(x).f(y) for all real numbers x and y. If f(3)=32 then f(1)=_

$${Given}\:{f}\left({x}+{y}\right)=\mathrm{4}{f}\left({x}\right).{f}\left({y}\right)\:{for} \\ $$$${all}\:{real}\:{numbers}\:{x}\:{and}\:{y}. \\ $$$${If}\:{f}\left(\mathrm{3}\right)=\mathrm{32}\:{then}\:{f}\left(\mathrm{1}\right)=\_ \\ $$

Question Number 119725 Answers: 3 Comments: 1

lim_(x→(π/4)) ((4(√2)−(cos x+sin x)^5 )/((cos x−sin x)^2 )) =?

$$\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\mathrm{4}\sqrt{\mathrm{2}}−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)^{\mathrm{5}} }{\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)^{\mathrm{2}} }\:=? \\ $$

Question Number 119724 Answers: 2 Comments: 0

Let x,y,z be non negative real numbers such that x+y+z=1. Find the extremum of F = 2x^2 +y+3z^2 .

$${Let}\:{x},{y},{z}\:{be}\:{non}\:{negative}\:{real}\:{numbers} \\ $$$${such}\:{that}\:{x}+{y}+{z}=\mathrm{1}.\:{Find}\:{the}\:{extremum} \\ $$$${of}\:{F}\:=\:\mathrm{2}{x}^{\mathrm{2}} +{y}+\mathrm{3}{z}^{\mathrm{2}} \:. \\ $$

Question Number 119713 Answers: 1 Comments: 3

If 3x+(1/(2x))=6 find 8x^3 +(1/(27x^3 ))

$$\mathrm{If}\:\mathrm{3x}+\frac{\mathrm{1}}{\mathrm{2x}}=\mathrm{6}\:\mathrm{find}\:\mathrm{8x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{27x}^{\mathrm{3}} } \\ $$

Question Number 119702 Answers: 0 Comments: 0

Question Number 119706 Answers: 0 Comments: 4

hello i hop all of you doing well can somoene help me too find lecture about dilogarithmes,trilogarithmes identities god bless you

$${hello}\:{i}\:{hop}\:{all}\:{of}\:{you}\:{doing}\:{well} \\ $$$${can}\:{somoene}\:{help}\:{me}\:{too}\:{find}\:{lecture} \\ $$$${about}\:{dilogarithmes},{trilogarithmes} \\ $$$${identities}\: \\ $$$${god}\:{bless}\:{you} \\ $$

Question Number 119696 Answers: 2 Comments: 0

For a<b then ∫_a ^b (x−a)(x−b) dx equal to _

$${For}\:{a}<{b}\:{then}\:\underset{{a}} {\overset{{b}} {\int}}\:\left({x}−{a}\right)\left({x}−{b}\right)\:{dx}\: \\ $$$${equal}\:{to}\:\_ \\ $$

Question Number 119692 Answers: 3 Comments: 1

If x is real number satisfying 3x+(1/(2x))=4 , find the value of 27x^3 +(1/(8x^3 )) .

$${If}\:{x}\:{is}\:{real}\:{number}\:{satisfying} \\ $$$$\mathrm{3}{x}+\frac{\mathrm{1}}{\mathrm{2}{x}}=\mathrm{4}\:,\:{find}\:{the}\:{value}\:{of} \\ $$$$\mathrm{27}{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{8}{x}^{\mathrm{3}} }\:. \\ $$

Question Number 119685 Answers: 0 Comments: 0

Question Number 119684 Answers: 1 Comments: 2

Question Number 119681 Answers: 4 Comments: 0

∫_0 ^π (√((1+cos2x)/2)) dx ∫_0 ^∞ [ne^(−x) ]dx

$$\int_{\mathrm{0}} ^{\pi} \sqrt{\frac{\mathrm{1}+{cos}\mathrm{2}{x}}{\mathrm{2}}}\:{dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} \left[{ne}^{−{x}} \right]{dx} \\ $$

Question Number 119679 Answers: 1 Comments: 0

2^1 ×2^2 ×2^3 ×...×2^n =(0.5)^(−45) n=?

$$\mathrm{2}^{\mathrm{1}} ×\mathrm{2}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{3}} ×...×\mathrm{2}^{{n}} =\left(\mathrm{0}.\mathrm{5}\right)^{−\mathrm{45}} \\ $$$${n}=? \\ $$

Question Number 119661 Answers: 0 Comments: 2

point A is at the bottom of a rough plane which is inclined at an angle Θ to the horizontal. A body of mass m is projected from A along and end up a line of greatest slope (along the plane). the cofficient of friction between the body and the plane is ϕ. it then comes to rest at point B at a distance X from A. obtain the expression for (a) the workdone against friction when the body moves from A to B and back to A (ii) initial speed of the body (iii) the speed of the body on its return to A

$${point}\:{A}\:{is}\:{at}\:{the}\:{bottom}\:{of}\:{a}\:{rough}\: \\ $$$${plane}\:{which}\:{is}\:{inclined}\:{at}\:{an}\:{angle}\: \\ $$$$\Theta\:{to}\:{the}\:{horizontal}.\:{A}\:{body}\:{of}\:{mass}\: \\ $$$${m}\:{is}\:{projected}\:{from}\:{A}\:{along}\:{and}\:{end}\: \\ $$$${up}\:{a}\:{line}\:{of}\:{greatest}\:{slope}\:\left({along}\:{the}\right. \\ $$$$\left.{plane}\right).\:{the}\:{cofficient}\:{of}\:{friction}\:{between} \\ $$$${the}\:{body}\:{and}\:{the}\:{plane}\:{is}\:\varphi.\:{it}\:{then}\: \\ $$$${comes}\:{to}\:{rest}\:{at}\:{point}\:{B}\:{at}\:{a}\:{distance}\:{X}\: \\ $$$${from}\:{A}.\:{obtain}\:{the}\:{expression}\:{for} \\ $$$$\left({a}\right)\:{the}\:{workdone}\:{against}\:{friction} \\ $$$${when}\:{the}\:{body}\:{moves}\:{from}\:{A}\:{to}\:{B} \\ $$$${and}\:{back}\:{to}\:{A} \\ $$$$\left({ii}\right)\:{initial}\:{speed}\:{of}\:{the}\:{body} \\ $$$$\left({iii}\right)\:{the}\:{speed}\:{of}\:{the}\:{body}\:{on}\:{its}\: \\ $$$${return}\:{to}\:{A} \\ $$

Question Number 119659 Answers: 2 Comments: 0

∫_0 ^1 [(4x−1)f(12x^2 −6x)]dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left[\left(\mathrm{4}{x}−\mathrm{1}\right){f}\left(\mathrm{12}{x}^{\mathrm{2}} −\mathrm{6}{x}\right)\right]{dx}=? \\ $$

Question Number 119657 Answers: 1 Comments: 2

Suppose that the greatest common divisor of the positive integers a,b and c is 1 and ((ab)/(a−b)) = c . Prove that a−b is a perfect square

$${Suppose}\:{that}\:{the}\:{greatest}\:{common}\:{divisor}\:{of} \\ $$$${the}\:{positive}\:{integers}\:{a},{b}\:{and}\:{c}\:{is}\:\mathrm{1}\:{and} \\ $$$$\frac{{ab}}{{a}−{b}}\:=\:{c}\:.\:{Prove}\:{that}\:{a}−{b}\:{is}\:{a} \\ $$$${perfect}\:{square} \\ $$

Question Number 119648 Answers: 1 Comments: 1

Particles of mass m_1 and m_2 (m_2 >m_1 ) are connected by a light inextensible string passing over a smooth fixed pulley. initially both masses hang vertically with mass m_(2 ) at a height X above the floor. if the system is released from rest. with what speed will mass m_2 hit the floor and the mass m_1 will rise a further distance of [(((m_2 −m_1 )x)/(m_1 +m_2 ))] after this occur.

$${Particles}\:{of}\:{mass}\:{m}_{\mathrm{1}} \:{and}\:{m}_{\mathrm{2}} \:\left({m}_{\mathrm{2}} >{m}_{\mathrm{1}} \right) \\ $$$${are}\:{connected}\:{by}\:{a}\:{light}\:{inextensible} \\ $$$${string}\:{passing}\:{over}\:{a}\:{smooth}\:{fixed}\: \\ $$$${pulley}.\:{initially}\:{both}\:{masses}\:{hang} \\ $$$${vertically}\:{with}\:{mass}\:{m}_{\mathrm{2}\:} {at}\:{a}\:{height} \\ $$$${X}\:{above}\:{the}\:{floor}.\:{if}\:{the}\:{system}\:{is}\: \\ $$$${released}\:{from}\:{rest}.\:{with}\:{what}\:{speed} \\ $$$${will}\:{mass}\:{m}_{\mathrm{2}} \:{hit}\:{the}\:{floor}\:{and}\:{the} \\ $$$${mass}\:{m}_{\mathrm{1}} \:{will}\:{rise}\:{a}\:{further}\:{distance}\:{of}\: \\ $$$$\left[\frac{\left({m}_{\mathrm{2}} −{m}_{\mathrm{1}} \right){x}}{{m}_{\mathrm{1}} +{m}_{\mathrm{2}} }\right]\:{after}\:{this}\:{occur}. \\ $$

Question Number 119647 Answers: 1 Comments: 0

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