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Question Number 220176 Answers: 0 Comments: 0

∫ (dx/(1 + x^7 ))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{{dx}}{\mathrm{1}\:+\:{x}^{\mathrm{7}} }\: \\ $$$$ \\ $$

Question Number 220164 Answers: 2 Comments: 0

if α^2 −5α+2=0 & β^2 −5β+2=0 then ((4α+β^5 )/(5β^2 ))=?

$${if}\:\:\alpha^{\mathrm{2}} −\mathrm{5}\alpha+\mathrm{2}=\mathrm{0}\:\:\&\:\:\beta^{\mathrm{2}} −\mathrm{5}\beta+\mathrm{2}=\mathrm{0} \\ $$$${then}\:\:\frac{\mathrm{4}\alpha+\beta^{\mathrm{5}} }{\mathrm{5}\beta^{\mathrm{2}} }=? \\ $$

Question Number 220160 Answers: 0 Comments: 0

for all x , y [0 , 1] ; prove that; [ (((x^3 + y^3 + 𝛇(3)))^(1/(3 )) /(1 + e^(−x^2 y^2 ) )) + (((x^4 + 𝚪(y+1)))^(1/(4 )) /((1 + y^2 )^(1/3) )) + ((ln(1 + x^5 + y^5 ))/( (√(1 + x^2 + y^2 )))) + Li_2 (xy) + ((√(x^6 + y^6 +1 ))/((1 + x^3 y^3 )^(1/2) )) ≤ (e^(xy) /(1 + x + y )) + ((ln (1 + x^2 + y^2 ) ))^(1/(3 )) + ((2𝛇(2))/( (√(1 + x^2 y^2 )))) + ((x^8 + y^8 + 1))^(1/(4 )) ]

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x}\:,\:{y}\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\left[\:\frac{\sqrt[{\mathrm{3}\:\:}]{\boldsymbol{{x}}^{\mathrm{3}} \:+\:\boldsymbol{{y}}^{\mathrm{3}} \:+\:\boldsymbol{\zeta}\left(\mathrm{3}\right)}}{\mathrm{1}\:+\:\boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} } \:}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{x}}^{\mathrm{4}} +\:\boldsymbol{\Gamma}\left(\boldsymbol{{y}}+\mathrm{1}\right)}}{\left(\mathrm{1}\:+\:\boldsymbol{{y}}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} }\:+\:\frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{5}} \:+\:\boldsymbol{{y}}^{\mathrm{5}} \right)}{\:\sqrt{\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} }}\:\:\:\:\:\:\:\:\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\boldsymbol{{xy}}\right)\:+\:\frac{\sqrt{\boldsymbol{{x}}^{\mathrm{6}} \:+\:\boldsymbol{{y}}^{\mathrm{6}} \:+\mathrm{1}\:}}{\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{3}} \boldsymbol{{y}}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{2}} }\: \\ $$$$\left.\:\:\:\:\:\:\leqslant\:\frac{\boldsymbol{{e}}^{\boldsymbol{{xy}}} }{\mathrm{1}\:+\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:}\:+\:\sqrt[{\mathrm{3}\:\:}]{\boldsymbol{\mathrm{ln}}\:\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} \right)\:}\:+\:\frac{\mathrm{2}\boldsymbol{\zeta}\left(\mathrm{2}\right)}{\:\sqrt{\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} }}\:+\:\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{x}}^{\mathrm{8}} \:+\:\boldsymbol{{y}}^{\mathrm{8}} \:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\right]\:\: \\ $$$$ \\ $$$$\:\: \\ $$

Question Number 220159 Answers: 0 Comments: 1

for all x, y ∈ [0 , 1] ; prove that; (1/( (√(1 + x^4 )))) + (2/( (√(1 + y^4 )))) + (2/( (√(4 + (x + y)^4 )))) + ((2(√2))/( (√(2+ x^2 y^2 + y^3 )))) ≤ (2/( (√(1 + x^2 y^2 )))) + (2/(^4 (√(1 + x^5 + y^5 )))) + ln(e+((x^3 y+y^3 x)/(1 + xy))) + (1/((1+x+y)^3 ))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x},\:{y}\:\in\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{y}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{4}\:+\:\left({x}\:+\:{y}\right)^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{3}} }}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\leqslant\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} }}\:+\:\frac{\mathrm{2}}{\:^{\mathrm{4}} \sqrt{\mathrm{1}\:+\:{x}^{\mathrm{5}} \:+\:{y}^{\mathrm{5}} }}\:+\:\mathrm{ln}\left({e}+\frac{{x}^{\mathrm{3}} {y}+{y}^{\mathrm{3}} {x}}{\mathrm{1}\:+\:{xy}}\right)\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}+{y}\right)^{\mathrm{3}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220141 Answers: 0 Comments: 0

Σ_(n=1) ^∞ Σ_(m=1) ^∞ (1/((n^2 +m^2 )^(3/2) ))=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}^{\mathrm{2}} +{m}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }=? \\ $$

Question Number 220131 Answers: 2 Comments: 0

If f(x,y)=(((x^2 +y^2 )^n )/(2n(2n−1)))+xφ((y/x))+Ψ((y/x)), then using Euler′s theorem on homogenous functions,show that x^2 ((δ^2 f)/(δx^2 ))+2xy((δ^2 f)/(δxδy))+y^2 ((δ^2 f)/(δy^2 ))=(x^2 +y^2 )^n

$${If}\:\:\:{f}\left({x},{y}\right)=\frac{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{{n}} }{\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right)}+{x}\phi\left(\frac{{y}}{{x}}\right)+\Psi\left(\frac{{y}}{{x}}\right), \\ $$$${then}\:{using}\:{Euler}'{s}\:{theorem}\:{on}\:{homogenous}\:{functions},{show}\:{that} \\ $$$${x}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{x}^{\mathrm{2}} }+\mathrm{2}{xy}\frac{\delta^{\mathrm{2}} {f}}{\delta{x}\delta{y}}+{y}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{y}^{\mathrm{2}} }=\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{{n}} \\ $$

Question Number 220121 Answers: 1 Comments: 0

Let H_h =p_(h+1) /p_h , p_h ∈P , p_1 =2 Π_(h=1) ^∞ H_h =?? (Π_(h=1) ^∞ H_h =(3/2)∙(5/3)∙(7/5).........)

$$\mathrm{Let}\:{H}_{{h}} ={p}_{{h}+\mathrm{1}} /{p}_{{h}} \:,\:{p}_{{h}} \in\mathbb{P}\:,\:{p}_{\mathrm{1}} =\mathrm{2} \\ $$$$\underset{{h}=\mathrm{1}} {\overset{\infty} {\prod}}\:{H}_{{h}} =??\:\left(\underset{{h}=\mathrm{1}} {\overset{\infty} {\prod}}\:{H}_{{h}} =\frac{\mathrm{3}}{\mathrm{2}}\centerdot\frac{\mathrm{5}}{\mathrm{3}}\centerdot\frac{\mathrm{7}}{\mathrm{5}}.........\right) \\ $$

Question Number 220321 Answers: 1 Comments: 2

Question Number 220115 Answers: 3 Comments: 0

Question Number 220114 Answers: 5 Comments: 0

Question Number 220113 Answers: 1 Comments: 0

Question Number 220111 Answers: 2 Comments: 1

Question Number 220110 Answers: 2 Comments: 0

Question Number 220108 Answers: 7 Comments: 0

Question Number 220104 Answers: 1 Comments: 0

Question Number 220103 Answers: 8 Comments: 0

Question Number 220101 Answers: 0 Comments: 0

Question Number 220097 Answers: 1 Comments: 0

x ∈ Q ; x ≠ 1 (7/(x − 1)) + (6/x) − (4/(x + 1)) + ((3x + 5)/(x^2 − 1)) = (1/x)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\:\in\:\mathbb{Q}\:\:\:;\:\:\:\:{x}\:\neq\:\mathrm{1} \\ $$$$\:\frac{\mathrm{7}}{{x}\:−\:\mathrm{1}}\:+\:\frac{\mathrm{6}}{{x}}\:−\:\frac{\mathrm{4}}{{x}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{3}{x}\:+\:\mathrm{5}}{{x}^{\mathrm{2}} \:−\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{{x}} \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 220096 Answers: 1 Comments: 0

let a, b, c, d, e is a positive real numbers and K = a + b + c + d + e +1 . prove that; Σ_(cyc) (1/(k−a)) < (1/4) ((((e^3 d^3 c))^(1/(4 )) /(c^(3/4) d^(1/2) e^(1/4) (√a))) + (((d^( 3) c^2 b))^(1/(4 )) /(d^( 3/4) c^(1/2) b^(1/4) (√e))) + (((c^3 b^2 a))^(1/(4 )) /(c^(3/4) b^(1/2) a^(1/4) (√d))) + (((b^3 a^2 e))^(1/(4 )) /(b^(3/4) a^(1/2) e^(1/4) (√c))) + (((a^3 e^2 d))^(1/(4 )) /(a^(3/4) e^(1/2) d^(1/4) (√b))))

$$ \\ $$$$\:\:\:\mathrm{let}\:{a},\:{b},\:{c},\:{d},\:{e}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{and} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{K}\:=\:{a}\:+\:{b}\:+\:{c}\:+\:{d}\:+\:{e}\:+\mathrm{1}\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\underset{\boldsymbol{{cyc}}} {\sum}\:\frac{\mathrm{1}}{\boldsymbol{{k}}−\boldsymbol{{a}}}\:<\:\frac{\mathrm{1}}{\mathrm{4}}\:\left(\frac{\sqrt[{\mathrm{4}\:\:\:\:}]{\boldsymbol{{e}}^{\mathrm{3}} \boldsymbol{{d}}^{\mathrm{3}} \boldsymbol{{c}}}}{\boldsymbol{{c}}^{\mathrm{3}/\mathrm{4}} \boldsymbol{{d}}^{\mathrm{1}/\mathrm{2}} \boldsymbol{{e}}^{\mathrm{1}/\mathrm{4}} \sqrt{\boldsymbol{{a}}}}\:+\:\frac{\sqrt[{\mathrm{4}\:\:\:}]{\boldsymbol{{d}}^{\:\mathrm{3}} \boldsymbol{{c}}^{\mathrm{2}} \boldsymbol{{b}}}}{\boldsymbol{{d}}^{\:\mathrm{3}/\mathrm{4}} \boldsymbol{{c}}^{\mathrm{1}/\mathrm{2}} \boldsymbol{{b}}^{\mathrm{1}/\mathrm{4}} \sqrt{\boldsymbol{{e}}}}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{c}}^{\mathrm{3}} \boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{a}}}}{\boldsymbol{{c}}^{\mathrm{3}/\mathrm{4}} \boldsymbol{{b}}^{\mathrm{1}/\mathrm{2}} \boldsymbol{{a}}^{\mathrm{1}/\mathrm{4}} \sqrt{\boldsymbol{{d}}}}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{b}}^{\mathrm{3}} \boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{e}}}}{\boldsymbol{{b}}^{\mathrm{3}/\mathrm{4}} \boldsymbol{{a}}^{\mathrm{1}/\mathrm{2}} \boldsymbol{{e}}^{\mathrm{1}/\mathrm{4}} \sqrt{\boldsymbol{{c}}}}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{a}}^{\mathrm{3}} \boldsymbol{{e}}^{\mathrm{2}} \boldsymbol{{d}}}}{\boldsymbol{{a}}^{\mathrm{3}/\mathrm{4}} \boldsymbol{{e}}^{\mathrm{1}/\mathrm{2}} \boldsymbol{{d}}^{\mathrm{1}/\mathrm{4}} \sqrt{\boldsymbol{{b}}}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$\:\: \\ $$

Question Number 220094 Answers: 1 Comments: 0

let n ≥ 2 ∈ Z and x_1 , x_2 , ..., x_n are a positive real numbers such that Σ_(i=1) ^n x_i = n , prove that Σ_(i=1) ^n (x_i ^n /(x_1 + ∙∙∙ + x_i ^ + ∙∙∙ + x_n )) ≥ (n/(n − 1))

$$ \\ $$$$\:\:\:\:\mathrm{let}\:{n}\:\geqslant\:\mathrm{2}\:\in\:\mathbb{Z}\:\mathrm{and}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:...,\:{x}_{{n}} \:\mathrm{are}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{such}\:\mathrm{that}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}_{{i}} \:=\:{n}\:,\:\mathrm{prove}\:\mathrm{that}\:\:\:\: \\ $$$$\:\:\:\:\:\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{x}_{{i}} ^{{n}} }{{x}_{\mathrm{1}} +\:\centerdot\centerdot\centerdot\:+\:\hat {{x}}_{{i}} \:+\:\centerdot\centerdot\centerdot\:+\:{x}_{{n}} }\:\geqslant\:\frac{{n}}{{n}\:−\:\mathrm{1}} \\ $$$$ \\ $$

Question Number 220081 Answers: 2 Comments: 3

Question Number 220074 Answers: 0 Comments: 0

Question Number 220072 Answers: 1 Comments: 0

If x,y∈(0,(π/2)) Then prove that: log_(sinx) ^2 (((sin2x)/(sinx + cosx))) + log_(cosx) ^2 (((sin2x)/(sinx + cosx))) ≥ 2

$$\mathrm{If}\:\:\:\mathrm{x},\mathrm{y}\in\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{log}_{\boldsymbol{\mathrm{sinx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:+\:\mathrm{log}_{\boldsymbol{\mathrm{cosx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:\geqslant\:\mathrm{2} \\ $$

Question Number 220069 Answers: 1 Comments: 0

Let be (H_n )_(n≥1) H_n = Σ_(k=1) ^n (1/k) Find: lim_(n→∞) e^(2H_n ) ((((n+1)!))^(1/(n+1)) − ((n!))^(1/n) ) sin (π/n^2 ) = ?

Question Number 220066 Answers: 0 Comments: 0

evaluate −((csc(πs))/(iπ))∫_( C) (−t)^(s−1) e^(−t) dt , path C;(−∞,∞) −((𝚪(1−s))/(2πi)) ∫_( C) (((−t)^(s−1) )/(e^t −1)) dt , path C;(−∞,∞)

$$\mathrm{evaluate} \\ $$$$−\frac{\mathrm{csc}\left(\pi{s}\right)}{\boldsymbol{{i}}\pi}\int_{\:\boldsymbol{\mathcal{C}}} \:\left(−{t}\right)^{{s}−\mathrm{1}} {e}^{−{t}} \:\mathrm{d}{t}\:,\:\mathrm{path}\:\boldsymbol{\mathcal{C}};\left(−\infty,\infty\right) \\ $$$$−\frac{\boldsymbol{\Gamma}\left(\mathrm{1}−{s}\right)}{\mathrm{2}\pi\boldsymbol{{i}}}\:\int_{\:\boldsymbol{\mathcal{C}}} \:\frac{\left(−{t}\right)^{{s}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}\:\mathrm{d}{t}\:,\:\mathrm{path}\:\boldsymbol{\mathcal{C}};\left(−\infty,\infty\right) \\ $$

Question Number 220065 Answers: 2 Comments: 0

α∈R ; ω∈R^+ I(α) = ∫_(−∞) ^( ∞) ((x^2 ln(1+x^2 ))/((x^4 +1)^α )) e^(−x^2 ) cos(ωx) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\alpha\in\mathbb{R}\:\:\:;\:\:\:\:\omega\in\mathbb{R}^{+} \\ $$$$\:\:\:\:\:{I}\left(\alpha\right)\:=\:\int_{−\infty} ^{\:\infty} \:\frac{{x}^{\mathrm{2}} \:\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\alpha} }\:{e}^{−{x}^{\mathrm{2}} } \:\mathrm{cos}\left(\omega{x}\right)\:{dx} \\ $$$$ \\ $$

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