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Question Number 144756    Answers: 1   Comments: 1

Question Number 144754    Answers: 0   Comments: 0

consider a random variable definite by geometric law compute P({X≥4})

$${consider}\:{a}\:{random}\:{variable}\:{definite}\:{by} \\ $$$${geometric}\:{law}\:{compute}\:{P}\left(\left\{{X}\geqslant\mathrm{4}\right\}\right) \\ $$

Question Number 144753    Answers: 1   Comments: 0

Question Number 144744    Answers: 1   Comments: 0

Question Number 144742    Answers: 0   Comments: 0

Let a,b,c>0 and a+b+c = 3. Prove that ((((ab)/(ab+1))+((bc)/(bc+1))+((ca)/(ca+1)))/((1/(ab+1))+(1/(bc+1))+(1/(ca+1)))) ≥ abc (Found by WolframAlpha and inspired by my old problem)

$$\mathrm{Let}\:{a},{b},{c}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\frac{{ab}}{{ab}+\mathrm{1}}+\frac{{bc}}{{bc}+\mathrm{1}}+\frac{{ca}}{{ca}+\mathrm{1}}}{\frac{\mathrm{1}}{{ab}+\mathrm{1}}+\frac{\mathrm{1}}{{bc}+\mathrm{1}}+\frac{\mathrm{1}}{{ca}+\mathrm{1}}}\:\geqslant\:{abc} \\ $$$$\left(\mathrm{Found}\:\mathrm{by}\:\mathrm{WolframAlpha}\:\mathrm{and}\:\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{inspired}\:\mathrm{by}\:\mathrm{my}\:\mathrm{old}\:\mathrm{problem}\right) \\ $$

Question Number 144738    Answers: 1   Comments: 0

On souhaite calculer I=∫_0 ^∞ ((sint)/t)dt. (1) On de^ finit la fonction F(x)=∫_0 ^∞ e^(−tx) ((sint)/t)dt. (a) De^ terminer le domaine de de^ finition de f sur R. (b) Montrer que F est de classe C^1 sur R_+ ^∗ et calculer F ′(x). (c) Limite de F en +∞ ? Conse^ quence ? (2) On note Si(t)=∫_0 ^t ((sinu)/u)du pour tout re^ el t. (a) Montrer que G(x)=∫_0 ^∞ e^(−tx) Si(t)dt est de^ finie sur R_+ ^∗ . (b) Montrer que xG(x)→I quand x→0^+ . (c) Au moyen d′une inte^ gration par parties, montrer que F est continue en 0. (3) Calculer I.

$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{On}\:\mathrm{souhaite}\:\mathrm{calculer}\:\mathrm{I}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{t}}{{t}}{dt}. \\ $$$$\left(\mathrm{1}\right)\:\mathrm{On}\:\mathrm{d}\acute {\mathrm{e}finit}\:\mathrm{la}\:\mathrm{fonction}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{tx}} \frac{\mathrm{sin}{t}}{{t}}{dt}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{a}\right)\:\mathrm{D}\acute {\mathrm{e}terminer}\:\mathrm{le}\:\mathrm{domaine}\:\mathrm{de}\:\mathrm{d}\acute {\mathrm{e}finition}\:\mathrm{de}\:{f}\:\mathrm{sur}\:\mathbb{R}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{Montrer}\:\mathrm{que}\:{F}\:\mathrm{est}\:\mathrm{de}\:\mathrm{classe}\:{C}^{\mathrm{1}} \:\mathrm{sur}\:{R}_{+} ^{\ast} \:\mathrm{et}\:\mathrm{calculer}\:{F}\:'\left({x}\right). \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{Limite}\:\mathrm{de}\:{F}\:\mathrm{en}\:+\infty\:?\:\mathrm{Cons}\acute {\mathrm{e}quence}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{On}\:\mathrm{note}\:{Si}\left({t}\right)=\int_{\mathrm{0}} ^{{t}} \frac{\mathrm{sin}{u}}{{u}}{du}\:\mathrm{pour}\:\mathrm{tout}\:\mathrm{r}\acute {\mathrm{e}el}\:{t}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{a}\right)\:\mathrm{Montrer}\:\mathrm{que}\:{G}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{tx}} {Si}\left({t}\right){dt}\:\mathrm{est}\:\mathrm{d}\acute {\mathrm{e}finie}\:\mathrm{sur}\:{R}_{+} ^{\ast} . \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{Montrer}\:\mathrm{que}\:{xG}\left({x}\right)\rightarrow{I}\:\mathrm{quand}\:{x}\rightarrow\mathrm{0}^{+} . \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{Au}\:\mathrm{moyen}\:\mathrm{d}'\mathrm{une}\:\mathrm{int}\acute {\mathrm{e}gration}\:\mathrm{par}\:\mathrm{parties},\:\mathrm{montrer}\:\mathrm{que}\:{F}\:\mathrm{est}\:\mathrm{continue}\:\mathrm{en}\:\mathrm{0}. \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Calculer}\:{I}. \\ $$

Question Number 144736    Answers: 3   Comments: 0

P(z)=az^3 +z^2 −(a+6)z+b−6 P(z)=(z^2 +4)∙Q(z) Find a∙b=?

$${P}\left({z}\right)={az}^{\mathrm{3}} +{z}^{\mathrm{2}} −\left({a}+\mathrm{6}\right){z}+{b}−\mathrm{6} \\ $$$${P}\left({z}\right)=\left({z}^{\mathrm{2}} +\mathrm{4}\right)\centerdot{Q}\left({z}\right) \\ $$$${Find}\:\:{a}\centerdot{b}=? \\ $$

Question Number 144734    Answers: 0   Comments: 0

Question Number 144733    Answers: 0   Comments: 0

Question Number 144727    Answers: 2   Comments: 0

Given { ((m=cos θ−sin θ)),((n=cos θ+sin θ)) :} then (√(m/n)) +(√(n/m)) = ?

$$\:\:\mathrm{Given}\:\begin{cases}{\mathrm{m}=\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}\\{\mathrm{n}=\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}\end{cases} \\ $$$$\:\:\mathrm{then}\:\sqrt{\frac{\mathrm{m}}{\mathrm{n}}}\:+\sqrt{\frac{\mathrm{n}}{\mathrm{m}}}\:=\:? \\ $$

Question Number 144724    Answers: 0   Comments: 0

Question Number 144721    Answers: 0   Comments: 0

......... Nice ......∗∗∗......Calculus......... f ( x ) : = [ tan (x) + cot (x) ] R_( f ) = ? Hint:: [ x ] := Max { m ∈Z ∣ m ≤ x }

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:.........\:\mathrm{Nice}\:......\ast\ast\ast......\mathrm{Calculus}......... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{f}\:\left(\:\mathrm{x}\:\right)\::\:=\:\left[\:\mathrm{tan}\:\left(\mathrm{x}\right)\:+\:\mathrm{cot}\:\left(\mathrm{x}\right)\:\right] \\ $$$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{R}_{\:\mathrm{f}\:\:} \:=\:? \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Hint}::\:\:\:\left[\:\mathrm{x}\:\right]\::=\:\mathrm{Max}\:\left\{\:\mathrm{m}\:\in\mathbb{Z}\:\mid\:\mathrm{m}\:\leqslant\:\mathrm{x}\:\right\}\: \\ $$

Question Number 144720    Answers: 1   Comments: 0

.....calculus..... Ω := ∫_0 ^( ∞) ((sech(πx))/(1+4x^( 2) )) dx =^? (1/2) Ln(2)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....\mathrm{calculus}..... \\ $$$$\: \\ $$$$\Omega\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sech}\left(\pi{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\:\mathrm{2}} }\:{dx}\:\overset{?} {=}\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{Ln}\left(\mathrm{2}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 144716    Answers: 1   Comments: 0

∫((dx/(e^(2x) +1)))=?

$$\int\left(\frac{{dx}}{{e}^{\mathrm{2}{x}} +\mathrm{1}}\right)=? \\ $$

Question Number 144708    Answers: 2   Comments: 0

Simplify: ((1/((x-2)!)) - (1/((x-1)!)))∙x!

$${Simplify}:\:\:\left(\frac{\mathrm{1}}{\left({x}-\mathrm{2}\right)!}\:-\:\frac{\mathrm{1}}{\left({x}-\mathrm{1}\right)!}\right)\centerdot{x}! \\ $$

Question Number 144705    Answers: 1   Comments: 0

∫ (x^(n−1) /(x^(3n+1) (x^n −a))) dx ?

$$\:\:\int\:\frac{\mathrm{x}^{\mathrm{n}−\mathrm{1}} }{\mathrm{x}^{\mathrm{3n}+\mathrm{1}} \:\left(\mathrm{x}^{\mathrm{n}} −\mathrm{a}\right)}\:\mathrm{dx}\:? \\ $$

Question Number 144702    Answers: 1   Comments: 0

let g(x)=log(cosx +2sinx) developp f at fourier serie

$$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)=\mathrm{log}\left(\mathrm{cosx}\:+\mathrm{2sinx}\right) \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$$$ \\ $$

Question Number 144701    Answers: 2   Comments: 0

calculate Σ_(n=1) ^∞ ((cos(nθ))/n^2 )

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{n}\theta\right)}{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 144700    Answers: 1   Comments: 0

find Σ_(n=1) ^∞ (x^n /n^2 )

$$\mathrm{find}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 144699    Answers: 3   Comments: 0

let f(x)=log(cht) developp f at fourier serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{log}\left(\mathrm{cht}\right) \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 144697    Answers: 2   Comments: 0

Evaluate (((1+cos (π/(10))−isin (π/(10)))/(1+cos (π/(10))+isin (π/(10)))))^(15) .

$$\mathrm{Evaluate}\:\left(\frac{\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{10}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{10}}}{\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{10}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{10}}}\right)^{\mathrm{15}} . \\ $$

Question Number 144693    Answers: 1   Comments: 0

Question Number 144691    Answers: 1   Comments: 0

Question Number 144684    Answers: 1   Comments: 0

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Question Number 144683    Answers: 1   Comments: 0

Determiner l′original de laplace F(p)=(1/((p^2 +p+1)^2 ))

$${Determiner}\:{l}'{original}\:{de}\:{laplace} \\ $$$${F}\left({p}\right)=\frac{\mathrm{1}}{\left({p}^{\mathrm{2}} +{p}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 144682    Answers: 1   Comments: 0

Compare: x=((sin(3))/(sin(5))) and y=((cos(3))/(cos(5)))

$${Compare}:\:\:{x}=\frac{{sin}\left(\mathrm{3}\right)}{{sin}\left(\mathrm{5}\right)}\:\:{and}\:\:{y}=\frac{{cos}\left(\mathrm{3}\right)}{{cos}\left(\mathrm{5}\right)} \\ $$

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