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Question Number 163658 Answers: 2 Comments: 0

lim_(x→∞) ((sin (π/(2n))×sin ((2π)/(2n))×sin ((3π)/(2n))....×sin (((n−1)π)/n)))^(1/n) =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\sqrt[{{n}}]{\mathrm{sin}\:\frac{\pi}{\mathrm{2}{n}}×\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{2}{n}}×\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{2}{n}}....×\mathrm{sin}\:\frac{\left({n}−\mathrm{1}\right)\pi}{{n}}}=? \\ $$

Question Number 163657 Answers: 2 Comments: 0

lim_(x→0) (((1+x)^(1/x) −e)/x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{{x}}} −{e}}{{x}}=? \\ $$

Question Number 163656 Answers: 1 Comments: 0

Question Number 163556 Answers: 2 Comments: 0

Question Number 163548 Answers: 0 Comments: 0

Lambert series type representation for (√(-1)) factorial (i^i )! (-i^i )! = 1 - 2 Σ_(n=1) ^∞ (((-1)^n )/(e^𝛑 n^2 - 1))

$$\mathrm{Lambert}\:\mathrm{series}\:\mathrm{type}\:\mathrm{representation} \\ $$$$\mathrm{for}\:\:\sqrt{-\mathrm{1}}\:\:\mathrm{factorial} \\ $$$$\left(\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:\left(-\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:=\:\mathrm{1}\:-\:\mathrm{2}\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} }{\mathrm{e}^{\boldsymbol{\pi}} \:\mathrm{n}^{\mathrm{2}} \:-\:\mathrm{1}} \\ $$

Question Number 163536 Answers: 2 Comments: 0

Prove that: ∫_( 9) ^( 1) ((ln (1 + x))/(1 + x^2 )) dx = ((π ∙ ln (2))/8)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{9}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:\frac{\pi\:\centerdot\:\mathrm{ln}\:\left(\mathrm{2}\right)}{\mathrm{8}} \\ $$

Question Number 163533 Answers: 0 Comments: 5

Re^ soudre (∂^2 u/∂x^2 )+(∂^2 u/∂y^2 )=e^(2x+y)

$$\mathrm{R}\acute {\mathrm{e}soudre}\:\:\:\:\:\:\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {u}}{\partial{y}^{\mathrm{2}} }={e}^{\mathrm{2}{x}+{y}} \\ $$

Question Number 163527 Answers: 0 Comments: 1

Question Number 163522 Answers: 3 Comments: 0

Question Number 163510 Answers: 0 Comments: 1

If 𝛗=∫_0 ^( (π/2)) (( 1)/( (√(sin^( 5) (x).cos(x))) +(√(cos^( 5) (x).sin(x)))))dx = find the value of : Γ^( 2) ((3/4) ). 𝛗

$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}{f} \\ $$$$\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:\mathrm{1}}{\:\sqrt{{sin}^{\:\mathrm{5}} \left({x}\right).{cos}\left({x}\right)}\:+\sqrt{{cos}^{\:\mathrm{5}} \left({x}\right).{sin}\left({x}\right)}}{dx}\:= \\ $$$$\:\:\:{find}\:{the}\:{value}\:{of}\:\::\:\Gamma^{\:\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\:\right).\:\boldsymbol{\phi} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 163508 Answers: 0 Comments: 0

etudier la continuite de [ x ] − (√(x − [ x ]))

$${etudier}\:{la}\:{continuite}\:{de}\:\left[\:{x}\:\right]\:−\:\sqrt{{x}\:−\:\left[\:{x}\:\right]} \\ $$

Question Number 163497 Answers: 1 Comments: 1

Question Number 163496 Answers: 1 Comments: 0

∫_0 ^(π/2) 256cos^5 ((x/2))sin^(11) ((x/2))dx

$$\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{256}\boldsymbol{{cos}}^{\mathrm{5}} \left(\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)\boldsymbol{{sin}}^{\mathrm{11}} \left(\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)\boldsymbol{{dx}} \\ $$

Question Number 163490 Answers: 1 Comments: 0

∫_0 ^∞ (x^3 /(e^x −1)) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{\boldsymbol{{x}}^{\mathrm{3}} }{\boldsymbol{{e}}^{\boldsymbol{{x}}} \:−\mathrm{1}}\:\boldsymbol{{dx}} \\ $$

Question Number 163487 Answers: 1 Comments: 0

Ω= ∫_0 ^( 1) (( sin^( 2) ( ln(x )). ln (x))/( (√x))) dx=? −−−−−

$$ \\ $$$$\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{sin}^{\:\mathrm{2}} \left(\:\mathrm{ln}\left({x}\:\right)\right).\:\mathrm{ln}\:\left({x}\right)}{\:\sqrt{{x}}}\:{dx}=? \\ $$$$\:\:\:\:−−−−− \\ $$

Question Number 163484 Answers: 0 Comments: 0

pour tout couple (a,b)εR^2 ,prouver que ∫_0 ^1 t^p (lnt)^q dt converge puis calculer

$${pour}\:{tout}\:{couple}\:\left({a},{b}\right)\epsilon{R}^{\mathrm{2}} ,{prouver}\:{que} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{p}} \left({lnt}\right)^{{q}} {dt}\:{converge}\:{puis}\:{calculer} \\ $$

Question Number 163483 Answers: 4 Comments: 0

Question Number 163481 Answers: 1 Comments: 0

Question Number 163473 Answers: 0 Comments: 2

Question Number 163472 Answers: 1 Comments: 0

∫_0 ^3 ((xdx)/(x^3 + 2x^2 + x 2)) =

$$\int_{\mathrm{0}} ^{\mathrm{3}} \:\frac{\boldsymbol{\mathrm{xdx}}}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:+\:\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\boldsymbol{\mathrm{x}}\:\mathrm{2}}\:= \\ $$

Question Number 163470 Answers: 0 Comments: 0

Question Number 163469 Answers: 1 Comments: 1

Question Number 163468 Answers: 0 Comments: 0

Question Number 163467 Answers: 1 Comments: 0

Question Number 163463 Answers: 1 Comments: 0

nature de la serie Σ_(n=1) ((1/(n+1))+(1/n))

$${nature}\:{de}\:{la}\:{serie} \\ $$$$\underset{{n}=\mathrm{1}} {\sum}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}+\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 163457 Answers: 0 Comments: 1

let a>0 and 𝛌>0 fixed solve for (0;∞) the equation: 2a^2 cos((x/(2λ)) - ((2λ)/x)) = a^(x/𝛌) + a^((4𝛌)/x)

$$\mathrm{let}\:\:\boldsymbol{\mathrm{a}}>\mathrm{0}\:\:\mathrm{and}\:\:\boldsymbol{\lambda}>\mathrm{0}\:\:\mathrm{fixed} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\:\left(\mathrm{0};\infty\right)\:\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{2a}^{\mathrm{2}} \mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}\lambda}\:-\:\frac{\mathrm{2}\lambda}{\mathrm{x}}\right)\:=\:\mathrm{a}^{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\lambda}}} \:\:+\:\:\mathrm{a}^{\frac{\mathrm{4}\boldsymbol{\lambda}}{\boldsymbol{\mathrm{x}}}} \\ $$

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