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Question Number 61386 Answers: 0 Comments: 3

find the value of ∫_(−∞) ^(+∞) ((ln(1+x^2 ))/(1+x^2 )) dx

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

Question Number 61378 Answers: 0 Comments: 0

Solve the differential equation: (dy/dx) = x^2 + y^2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\:=\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \\ $$

Question Number 61377 Answers: 0 Comments: 0

Find the multinomial coefficient: ((( 9)),((3, 5, 1, 0)) )

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{multinomial}\:\mathrm{coefficient}:\:\:\:\:\:\:\begin{pmatrix}{\:\:\:\:\:\:\:\mathrm{9}}\\{\mathrm{3},\:\mathrm{5},\:\mathrm{1},\:\mathrm{0}}\end{pmatrix} \\ $$

Question Number 61363 Answers: 1 Comments: 0

Question Number 61349 Answers: 1 Comments: 7

Question Number 61347 Answers: 2 Comments: 1

Question Number 61343 Answers: 1 Comments: 0

Question Number 61335 Answers: 1 Comments: 0

Question Number 61329 Answers: 0 Comments: 2

let f(x) =(e^(−x) /(1+x)) sin(3x) 1) dtermine f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie .

$${let}\:{f}\left({x}\right)\:=\frac{{e}^{−{x}} }{\mathrm{1}+{x}}\:{sin}\left(\mathrm{3}{x}\right) \\ $$$$\left.\mathrm{1}\right)\:{dtermine}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 61328 Answers: 0 Comments: 4

let f(a) =∫_0 ^1 ((sin(2x))/(1+ax^2 )) dx with ∣a∣<1 1) approximate f(a) by a polynom 2) find the value (perhaps not exact) of ∫_0 ^1 ((sin(2x))/(1+2x^2 )) dx 3) let g(a) = ∫_0 ^1 ((x^2 sin(2x))/((1+ax^2 )^2 )) dx approximat g(a) by a polynom 4) find the value of ∫_0 ^1 ((x^2 sin(2x))/((1+2x^2 )^2 )) dx .

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{ax}^{\mathrm{2}} }\:{dx}\:\:{with}\:\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{approximate}\:{f}\left({a}\right)\:{by}\:{a}\:{polynom} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:\:\left({perhaps}\:{not}\:{exact}\right)\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} {sin}\left(\mathrm{2}{x}\right)}{\left(\mathrm{1}+{ax}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:\:\:{approximat}\:{g}\left({a}\right)\:{by}\:{a}\:{polynom} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} {sin}\left(\mathrm{2}{x}\right)}{\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 61327 Answers: 1 Comments: 0

Question Number 61326 Answers: 0 Comments: 4

find ∫_0 ^1 ((sinx)/(1+x^2 ))dx

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sinx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 61322 Answers: 1 Comments: 0

Question Number 61320 Answers: 1 Comments: 0

solve for x: x^4 −x=12

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}: \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{4}} −\boldsymbol{\mathrm{x}}=\mathrm{12} \\ $$

Question Number 61318 Answers: 1 Comments: 0

Question Number 61313 Answers: 0 Comments: 3

how to calculate lim_(n→+∞) 0.05^(0.05^(0.05^.^.^(.0.05) ) ) }n in a simple and fast way?

$$\left.{how}\:{to}\:{calculate}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}0}.\mathrm{05}^{\mathrm{0}.\mathrm{05}^{\mathrm{0}.\mathrm{05}^{.^{.^{.\mathrm{0}.\mathrm{05}} } } } } \right\}{n} \\ $$$${in}\:{a}\:{simple}\:{and}\:{fast}\:{way}? \\ $$

Question Number 61303 Answers: 2 Comments: 3

Question Number 61297 Answers: 0 Comments: 0

A container is 50% full of water at triple point phase . It′s Isolated and subjected to space system defining no gravity acting on the particles , Which state of matter is now more dominant , solid , liquid , or gas ? Calculate the intermolecular distances between simultaneous two distinct states of water.

$$\mathrm{A}\:\mathrm{container}\:\mathrm{is}\:\mathrm{50\%}\:\mathrm{full}\:\mathrm{of}\:\mathrm{water}\:\mathrm{at}\:\mathrm{triple}\:\mathrm{point} \\ $$$$\mathrm{phase}\:.\:\mathrm{It}'\mathrm{s}\:\mathrm{Isolated}\:\mathrm{and}\:\mathrm{subjected}\:\mathrm{to}\:\mathrm{space} \\ $$$$\mathrm{system}\:\mathrm{defining}\:\mathrm{no}\:\mathrm{gravity}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{particles}\:,\:\mathrm{Which}\:\mathrm{state}\:\mathrm{of}\:\mathrm{matter}\:\mathrm{is}\:\mathrm{now} \\ $$$$\mathrm{more}\:\mathrm{dominant}\:,\:\mathrm{solid}\:,\:\mathrm{liquid}\:,\:\mathrm{or}\:\mathrm{gas}\:? \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{intermolecular}\:\mathrm{distances} \\ $$$$\mathrm{between}\:\mathrm{simultaneous}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{states} \\ $$$$\mathrm{of}\:\mathrm{water}.\: \\ $$

Question Number 61284 Answers: 0 Comments: 0

(998^(999) × 999^(998) × 2019^(2019) ) mod (1000) = ?

$$\left(\mathrm{998}^{\mathrm{999}} \:×\:\mathrm{999}^{\mathrm{998}} \:×\:\mathrm{2019}^{\mathrm{2019}} \right)\:\:{mod}\:\left(\mathrm{1000}\right)\:\:=\:\:? \\ $$

Question Number 61283 Answers: 1 Comments: 0

(x+y)(x^2 +y^2 )(x^3 + y^3 ) = 2 (x^4 +y^4 )(x^6 +y^6 )(x^8 + y^8 ) = 4 (x^3 + y^3 )(x^5 + y^5 )(x^7 + y^7 ) = 6 (x^4 + y^4 )(x^5 + y^5 )(x^9 + y^9 )(x^(10) + y^(10) ) = ?

$$\left({x}+{y}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\left({x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \right)\:\:=\:\:\mathrm{2} \\ $$$$\left({x}^{\mathrm{4}} +{y}^{\mathrm{4}} \right)\left({x}^{\mathrm{6}} +{y}^{\mathrm{6}} \right)\left({x}^{\mathrm{8}} \:+\:{y}^{\mathrm{8}} \right)\:\:=\:\:\mathrm{4} \\ $$$$\left({x}^{\mathrm{3}} +\:{y}^{\mathrm{3}} \right)\left({x}^{\mathrm{5}} +\:{y}^{\mathrm{5}} \right)\left({x}^{\mathrm{7}} \:+\:{y}^{\mathrm{7}} \right)\:\:=\:\:\mathrm{6} \\ $$$$\left({x}^{\mathrm{4}} \:+\:{y}^{\mathrm{4}} \right)\left({x}^{\mathrm{5}} \:+\:{y}^{\mathrm{5}} \right)\left({x}^{\mathrm{9}} \:+\:{y}^{\mathrm{9}} \right)\left({x}^{\mathrm{10}} \:+\:{y}^{\mathrm{10}} \right)\:\:=\:\:? \\ $$

Question Number 61274 Answers: 1 Comments: 0

If p , x_1 ,x_2 ,...x_i and q,y_1 ,y_2 ,...y_i form two infinite arithmetic sequences with common difference a and b respectively , then find the locus of the point ( α , β ) where α = (1/n) Σ_(i=1) ^n x_i and β= (1/n) Σ_(i=1) ^n y_(i .)

$${If}\:{p}\:,\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...{x}_{{i}} \:{and}\:{q},{y}_{\mathrm{1}} ,{y}_{\mathrm{2}} ,...{y}_{{i}} \:{form}\:{two}\: \\ $$$${infinite}\:{arithmetic}\:{sequences}\:{with}\:{common}\: \\ $$$${difference}\:\:{a}\:{and}\:{b}\:{respectively}\:, \\ $$$${then}\:{find}\:{the}\:{locus}\:{of}\:{the}\:{point}\:\left(\:\alpha\:,\:\beta\:\right)\: \\ $$$${where}\:\alpha\:=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {x}_{{i}} \:{and}\:\beta=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {y}_{{i}\:.} \\ $$

Question Number 61272 Answers: 0 Comments: 0

Question Number 61273 Answers: 2 Comments: 2

Suppose α ,β,γ,δ are real numbers such that α+β+γ+δ = α^7 +β^7 +γ^7 +δ^7 =0 Prove that α(α+β)(α+γ)(α+δ)=0

$${Suppose}\:\alpha\:,\beta,\gamma,\delta\:{are}\:{real}\:{numbers} \\ $$$${such}\:{that}\:\alpha+\beta+\gamma+\delta\:=\:\alpha^{\mathrm{7}} +\beta^{\mathrm{7}} +\gamma^{\mathrm{7}} +\delta^{\mathrm{7}} =\mathrm{0} \\ $$$${Prove}\:{that}\:\alpha\left(\alpha+\beta\right)\left(\alpha+\gamma\right)\left(\alpha+\delta\right)=\mathrm{0} \\ $$

Question Number 61270 Answers: 2 Comments: 0

Question Number 61269 Answers: 3 Comments: 0

Let p(x) be a quadratic polynomial such that for distinct α and β , p(α) = α and p(β) =β prove that α and β are roots of p[p(x)]−x=0 Find the remaining roots .

$${Let}\:{p}\left({x}\right)\:{be}\:{a}\:{quadratic}\:{polynomial}\:{such} \\ $$$${that}\:{for}\:{distinct}\:\alpha\:{and}\:\beta\:, \\ $$$${p}\left(\alpha\right)\:=\:\alpha\:{and}\:{p}\left(\beta\right)\:=\beta \\ $$$${prove}\:{that}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\:\:{p}\left[{p}\left({x}\right)\right]−{x}=\mathrm{0}\: \\ $$$${Find}\:{the}\:{remaining}\:{roots}\:. \\ $$

Question Number 61268 Answers: 0 Comments: 0

Let a,b,c,d,e ≥ −1 and a+b+c+d+e=5 Find the maximum and minimum value of S =(a+b)(b+c)(c+d)(d+e)(e+a)

$${Let}\:{a},{b},{c},{d},{e}\:\geqslant\:−\mathrm{1}\:{and}\:{a}+{b}+{c}+{d}+{e}=\mathrm{5} \\ $$$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${value}\:{of}\:{S}\:=\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{d}\right)\left({d}+{e}\right)\left({e}+{a}\right) \\ $$

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