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Question Number 162171 Answers: 0 Comments: 0

Question Number 162169 Answers: 2 Comments: 0

determinant ((( determinant ((( x+y+x^2 y^2 =586_(x=?,y=? ) ^(x,y∈Z ) )))_ ^ _() ^(•) )))

$$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$

Question Number 162168 Answers: 0 Comments: 0

Solve the integro−differential equation: i(t) + 4(di/dt) + ∫i(t)dt = 2 cos (3t+ 60°) where i(t) is a sinulsodial current.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{integro}−\mathrm{differential} \\ $$$$\mathrm{equation}: \\ $$$$\:{i}\left({t}\right)\:+\:\mathrm{4}\frac{{di}}{{dt}}\:+\:\int{i}\left({t}\right){dt}\:=\:\mathrm{2}\:\mathrm{cos}\:\left(\mathrm{3}{t}+\:\mathrm{60}°\right) \\ $$$$\mathrm{where}\:{i}\left({t}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{sinulsodial}\:\mathrm{current}. \\ $$

Question Number 162151 Answers: 1 Comments: 0

Question Number 162116 Answers: 2 Comments: 0

A function f is such that f : R → R where f(xy+1) = f(x)∙f(y)−f(y)−x+2 , ∀x,y ∈ R . Find value of 10∙f(2017)+f(0) .

$${A}\:{function}\:\:{f}\:\:\:{is}\:\:{such}\:\:{that}\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\:{where} \\ $$$$\:\:\:{f}\left({xy}+\mathrm{1}\right)\:=\:{f}\left({x}\right)\centerdot{f}\left({y}\right)−{f}\left({y}\right)−{x}+\mathrm{2}\:\:,\:\:\forall{x},{y}\:\in\:\mathbb{R}\:. \\ $$$${Find}\:\:{value}\:\:{of}\:\:\mathrm{10}\centerdot{f}\left(\mathrm{2017}\right)+{f}\left(\mathrm{0}\right)\:. \\ $$

Question Number 162112 Answers: 1 Comments: 0

∫(( cos(x))/((1−cos(x))^2 ))dx=?

$$\int\frac{\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$

Question Number 162109 Answers: 1 Comments: 0

Question Number 162103 Answers: 1 Comments: 0

Find: lim_(x→0) ((sinx - sin^(-1) x)/(sinhx - sinh^(-1) x)) = ?

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sinx}\:-\:\mathrm{sin}^{-\mathrm{1}} \mathrm{x}}{\mathrm{sinhx}\:-\:\mathrm{sinh}^{-\mathrm{1}} \mathrm{x}}\:=\:? \\ $$

Question Number 162102 Answers: 0 Comments: 2

Solve for real numbers: 5^x + 4^(1/x) + 25^x ∙ 16^(1/x) = 2527

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{5}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{4}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:+\:\mathrm{25}^{\boldsymbol{\mathrm{x}}} \:\centerdot\:\mathrm{16}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:=\:\mathrm{2527} \\ $$

Question Number 162100 Answers: 1 Comments: 1

Question Number 162099 Answers: 0 Comments: 0

PROVE THAT Ω= ∫_0 ^( 1) (( Li_( 2) (x ). ln( x ))/x) dx =^? (( −π^( 4) )/(90)) −−−−−−−−−− Ω= ∫_0 ^( 1) ln (x )Σ_(n=1) ^∞ (( x^( n−1) )/n^( 2) ) dx = Σ_(n=1) ^∞ (1/n^( 2) ) ∫_0 ^( 1) x^( n−1) . ln(x ) dx = Σ_(n=1) ^∞ (1/n^( 2) ) {[ (x^( n) /n) ln( x )]_0 ^( 1) −(1/n) ∫_0 ^( 1) x^( n−1) dx} = Σ_(n=1) ^∞ ((−1)/n^( 4) ) = − ζ (4 ) = ((−π^( 4) )/( 90)) ■ m.n −−− M . N −−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{PROVE}\:\:\:\mathscr{THAT}\:\: \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right).\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:−\pi^{\:\mathrm{4}} }{\mathrm{90}} \\ $$$$\:\:\:\:\:−−−−−−−−−− \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\:\left({x}\:\right)\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{x}^{\:{n}−\mathrm{1}} }{{n}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}−\mathrm{1}} .\:\mathrm{ln}\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\left\{\left[\:\frac{{x}^{\:{n}} }{{n}}\:\mathrm{ln}\left(\:{x}\:\right)\right]_{\mathrm{0}} ^{\:\mathrm{1}} −\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {dx}\right\} \\ $$$$\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{−\mathrm{1}}{{n}^{\:\mathrm{4}} }\:=\:−\:\zeta\:\left(\mathrm{4}\:\right)\:=\:\frac{−\pi^{\:\mathrm{4}} }{\:\mathrm{90}}\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:−−−\:\mathscr{M}\:.\:\mathscr{N}\:\:−−−\: \\ $$$$ \\ $$

Question Number 162139 Answers: 2 Comments: 0

Question Number 162138 Answers: 1 Comments: 0

Question Number 162118 Answers: 0 Comments: 7

determiner le reste de la division euclidienne de 10^(100) par 105

$${determiner}\:{le}\:{reste}\:{de}\:{la}\:{division}\:{euclidienne}\:{de} \\ $$$$\mathrm{10}^{\mathrm{100}} \:{par}\:\mathrm{105} \\ $$

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

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

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