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AllQuestion and Answers: Page 505

Question Number 162117 Answers: 2 Comments: 1

Question Number 162092 Answers: 0 Comments: 0

Question Number 162088 Answers: 2 Comments: 0

x^2 =2^x solve for x=?

$${x}^{\mathrm{2}} =\mathrm{2}^{{x}} \\ $$$${solve}\:\:\:{for}\:\:\:\:{x}=? \\ $$

Question Number 162083 Answers: 0 Comments: 0

Question Number 162081 Answers: 2 Comments: 0

lim_(x→0) ((√((1/x)+(√((1/x)+(√(1/x)))))) −(√((1/x)−(√((1/x)+(√(1/x)))))) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:−\sqrt{\frac{\mathrm{1}}{{x}}−\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:=?\right. \\ $$

Question Number 162165 Answers: 2 Comments: 1

If the ratio of the roots of equation ax^2 +2bx+c=0 is same as the ratio of the roots of px^2 +2qx+r=0 where a,b,c,p ,r are non zero real numbers . Then the value of ((b^2 /q^2 ))((p/a))((r/c)) is equal to

$$\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:{ax}^{\mathrm{2}} +\mathrm{2}{bx}+{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{same}\:\mathrm{as}\:\mathrm{the}\:\mathrm{ratio} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{px}^{\mathrm{2}} +\mathrm{2}{qx}+{r}=\mathrm{0}\:\mathrm{where} \\ $$$$\:{a},{b},{c},{p}\:,{r}\:\mathrm{are}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left(\frac{{b}^{\mathrm{2}} }{{q}^{\mathrm{2}} }\right)\left(\frac{{p}}{{a}}\right)\left(\frac{{r}}{{c}}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\: \\ $$

Question Number 162074 Answers: 1 Comments: 0

Find the exact value of Σ_(k=0) ^(1004) ( _k ^(2014) ) ∙ 3^k .

$${Find}\:\:{the}\:\:{exact}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1004}} {\sum}}\:\left(\underset{{k}} {\overset{\mathrm{2014}} {\:}}\right)\:\centerdot\:\mathrm{3}^{{k}} \:. \\ $$

Question Number 162071 Answers: 1 Comments: 9

Question Number 162068 Answers: 1 Comments: 0

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))))

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

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