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Question Number 162066 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (((−1)^n H_n )/n^2 )=???

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=??? \\ $$

Question Number 162062 Answers: 1 Comments: 0

Ω(α;β) =∫_( -1) ^( 1) (((1+x)^(2𝛂-1) (1-x)^(2𝛃-1) )/((1+x^2 )^(𝛂+𝛃) )) dx ; α;β>0 find a closed form and prove that: Ω(3,5) > (√(Ω(4,5)∙Ω(3,6)))

$$\Omega\left(\alpha;\beta\right)\:=\underset{\:-\mathrm{1}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}\boldsymbol{\alpha}-\mathrm{1}} \:\left(\mathrm{1}-\mathrm{x}\right)^{\mathrm{2}\boldsymbol{\beta}-\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\boldsymbol{\alpha}+\boldsymbol{\beta}} }\:\mathrm{dx}\:;\:\alpha;\beta>\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{form}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\Omega\left(\mathrm{3},\mathrm{5}\right)\:>\:\sqrt{\Omega\left(\mathrm{4},\mathrm{5}\right)\centerdot\Omega\left(\mathrm{3},\mathrm{6}\right)} \\ $$

Question Number 162055 Answers: 1 Comments: 0

∫e^(2x) (√((1 −e^(2x) )))dx

$$\int{e}^{\mathrm{2x}} \sqrt{\left(\mathrm{1}\:−{e}^{\mathrm{2}{x}} \right)}{dx} \\ $$

Question Number 162054 Answers: 1 Comments: 0

Question Number 162043 Answers: 1 Comments: 2

Find valu of x if x∈R ((9x - 1))^(1/3) + (√(8x - 1)) + ((8x + 15))^(1/4) - (5/2) = 0

$$\mathrm{Find}\:\mathrm{valu}\:\mathrm{of}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{if}\:\:\mathrm{x}\in\mathbb{R}\: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{9x}\:-\:\mathrm{1}}\:+\:\sqrt{\mathrm{8x}\:-\:\mathrm{1}}\:+\:\sqrt[{\mathrm{4}}]{\mathrm{8x}\:+\:\mathrm{15}}\:-\:\frac{\mathrm{5}}{\mathrm{2}}\:=\:\mathrm{0} \\ $$

Question Number 162042 Answers: 0 Comments: 0

let a;b;c∈R such that a+b+c=3 prove that: a^3 + b^3 + c^3 ≥ a^3 b + b^3 c + c^3 a

$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\in\mathbb{R}\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \:\geqslant\:\mathrm{a}^{\mathrm{3}} \mathrm{b}\:+\:\mathrm{b}^{\mathrm{3}} \mathrm{c}\:+\:\mathrm{c}^{\mathrm{3}} \mathrm{a} \\ $$

Question Number 162073 Answers: 3 Comments: 0

prove that Ω =∫_(−∞) ^( +∞) (( cos (x))/((2+ 2x +x^( 2) )^( 2) )) dx = (π/e) cos(1)

$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\Omega\:=\int_{−\infty} ^{\:+\infty} \frac{\:{cos}\:\left({x}\right)}{\left(\mathrm{2}+\:\mathrm{2}{x}\:+{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{{e}}\:{cos}\left(\mathrm{1}\right) \\ $$

Question Number 162035 Answers: 2 Comments: 0

( _1 ^(2014) ) + ( _2 ^(2014) ) + ( _3 ^(2014) ) + …+ ( _(1007) ^(2014) ) = ?

$$\left(\underset{\mathrm{1}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\left(\underset{\mathrm{2}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\left(\underset{\mathrm{3}} {\overset{\mathrm{2014}} {\:}}\right)\:+\:\ldots+\:\left(\underset{\mathrm{1007}} {\overset{\mathrm{2014}} {\:}}\right)\:=\:? \\ $$

Question Number 162033 Answers: 0 Comments: 2

Question Number 162025 Answers: 0 Comments: 0

write the taylor expansion of : f(x)= x^( 2) . cos(x) at x=1 then f^( (5 )) (x) at x=1 ?

$$\:\:\:\: \\ $$$$\:\:\:{write}\:\:{the}\:{taylor}\:{expansion}\:{of}\:: \\ $$$$\:\:\:\:\:\:{f}\left({x}\right)=\:{x}^{\:\mathrm{2}} .\:{cos}\left({x}\right)\:\:\:\:{at}\:\:{x}=\mathrm{1} \\ $$$$\:\:\:\:{then}\:\:\:\:\:\:\:\:{f}^{\:\left(\mathrm{5}\:\right)} \left({x}\right)\:\:{at}\:\:{x}=\mathrm{1}\:\:? \\ $$$$ \\ $$

Question Number 162026 Answers: 2 Comments: 0

prove that.... ( 1+ (1/n) )^( n) < e < (1+(1/n) )^( n+1)

$$ \\ $$$$\:\:\:\:{prove}\:{that}.... \\ $$$$\: \\ $$$$\:\:\:\:\:\left(\:\mathrm{1}+\:\frac{\mathrm{1}}{{n}}\:\right)^{\:{n}} \:<\:{e}\:<\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\:\right)^{\:{n}+\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$

Question Number 162016 Answers: 2 Comments: 2

calculate ∫_(−∞) ^(+∞) ((cos(3x))/((x^2 +x+1)^2 ))dx

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{3x}\right)}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 162015 Answers: 2 Comments: 0

find ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)^4 ))

$$\mathrm{find}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 162003 Answers: 2 Comments: 0

((27((27((27.... ))^(1/4) ))^(1/4) ))^(1/4) =x (√(5(√(5(√(5(√(5.....))))))))=y y^2 −x^2 =?

$$\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}....\:}}}={x} \\ $$$$\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}.....}}}}={y} \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} =? \\ $$

Question Number 162002 Answers: 1 Comments: 0

calculate Ω = ∫_0 ^( 1) Li_( 2) (1 − x^( 4) )dx = ? −−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:{calculate} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{Li}_{\:\mathrm{2}} \:\left(\mathrm{1}\:−\:{x}^{\:\mathrm{4}} \right){dx}\:=\:? \\ $$$$\:\:\:\:−−−−− \\ $$

Question Number 162001 Answers: 1 Comments: 0

nature of: ∫_0 ^(+oo) ((sint)/(e^t −1))dt

$${nature}\:{of}: \\ $$$$\int_{\mathrm{0}} ^{+{oo}} \frac{{sint}}{{e}^{{t}} −\mathrm{1}}{dt} \\ $$

Question Number 161999 Answers: 1 Comments: 0

∫(dx/( (√(x^3 −4x))))

$$\int\frac{{dx}}{\:\sqrt{{x}^{\mathrm{3}} −\mathrm{4}{x}}} \\ $$

Question Number 161994 Answers: 0 Comments: 0

∫_0 ^1 ((ln∣x∣ln∣((1+x)/(1−x))∣)/(1−x^2 ))dx=???

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}\mid\boldsymbol{{x}}\mid\boldsymbol{\mathrm{ln}}\mid\frac{\mathrm{1}+\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}}\mid}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=??? \\ $$

Question Number 161991 Answers: 1 Comments: 1

Question Number 161968 Answers: 1 Comments: 0

Find coefficient of x^(29) in expansion of (1+x^5 +x^7 +x^9 )^(1000) .

$${Find}\:\:{coefficient}\:\:{of}\:\:{x}^{\mathrm{29}} \:\:{in}\:\:{expansion}\:\:{of}\:\:\:\left(\mathrm{1}+{x}^{\mathrm{5}} +{x}^{\mathrm{7}} +{x}^{\mathrm{9}} \right)^{\mathrm{1000}} \:. \\ $$

Question Number 161967 Answers: 3 Comments: 0

∫_0 ^1 ((ln^2 (x))/((1−x^2 )))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}\right)}{\left(\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 161966 Answers: 1 Comments: 0

∫x^2 7^x^2 dx=?

$$\int\boldsymbol{\mathrm{x}}^{\mathrm{2}} \mathrm{7}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 161964 Answers: 2 Comments: 0

Prove that: (a series inspired Knopp Konrad) (√e^𝛑 ) = Σ_(k=0) ^∞ ((sin(((kπ)/4)))/((k!) (√2^k ))) π^k

$$\mathrm{Prove}\:\mathrm{that}:\:\left(\mathrm{a}\:\mathrm{series}\:\mathrm{inspired}\:\mathrm{Knopp}\:\mathrm{Konrad}\right) \\ $$$$\sqrt{\mathrm{e}^{\boldsymbol{\pi}} }\:\:=\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{sin}\left(\frac{{k}\pi}{\mathrm{4}}\right)}{\left(\mathrm{k}!\right)\:\sqrt{\mathrm{2}^{\boldsymbol{\mathrm{k}}} }}\:\:\pi^{\boldsymbol{\mathrm{k}}} \\ $$

Question Number 161952 Answers: 0 Comments: 5

!!8=?

$$!!\mathrm{8}=? \\ $$

Question Number 161951 Answers: 2 Comments: 0

x^x =2^(2048) x=?

$${x}^{{x}} =\mathrm{2}^{\mathrm{2048}} \\ $$$${x}=? \\ $$

Question Number 161947 Answers: 2 Comments: 0

abc=8 a+b+c=7 a^3 +b^3 +c^3 =73 then faind the vole of (1/a)+(1/b)+(1/c)=?

$${abc}=\mathrm{8} \\ $$$${a}+{b}+{c}=\mathrm{7} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{73} \\ $$$${then}\:{faind}\:\:{the}\:{vole}\:{of} \\ $$$$\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}=? \\ $$

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