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Question Number 163111    Answers: 2   Comments: 0

calcul en fonction de n Σ_(k=0) ^n 3^(k−1) (_k ^n ) Σ_(k=0) ^(k=n) sin(kx)(_k ^n )

$${calcul}\:{en}\:{fonction}\:{de}\:{n} \\ $$$$\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{3}^{{k}−\mathrm{1}} \left(_{{k}} ^{{n}} \right) \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{k}={n}} {\sum}}{sin}\left({kx}\right)\left(_{{k}} ^{{n}} \right) \\ $$$$ \\ $$

Question Number 163098    Answers: 2   Comments: 0

Question Number 163095    Answers: 0   Comments: 2

what the best math′s app for android and pc?

$${what}\:{the}\:{best}\:{math}'{s}\:{app}\:{for}\:{android}\:{and}\:{pc}? \\ $$

Question Number 163082    Answers: 0   Comments: 3

1. u_(x x) - 2u_(x y) + 3u_(y y) - u_x = 0 2. xu_(x x) + 2xu_(x y) + 3u_(y y) + u_y = 0

$$\mathrm{1}.\:\mathrm{u}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{x}}} \:-\:\mathrm{2u}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{y}}} \:+\:\mathrm{3u}_{\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{y}}} \:-\:\mathrm{u}_{\boldsymbol{\mathrm{x}}} \:=\:\mathrm{0} \\ $$$$\mathrm{2}.\:\mathrm{xu}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2xu}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{y}}} \:+\:\mathrm{3u}_{\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{y}}} \:+\:\mathrm{u}_{\boldsymbol{\mathrm{y}}} \:=\:\mathrm{0} \\ $$

Question Number 163109    Answers: 1   Comments: 0

find the value of ∫_(−∞) ^(+∞) ((cosx)/((x^2 +1)^n ))dx (n fromN and n≥1)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cosx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{n}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{n}\:\mathrm{fromN}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{1}\right) \\ $$

Question Number 163080    Answers: 1   Comments: 0

F(x)= (((x^2 −4x)^2 ))^(1/3) {: ((local maximum)),((absolut maximum)) } =?

$$\:\:\:\:\:\:\:{F}\left({x}\right)=\:\sqrt[{\mathrm{3}}]{\left({x}^{\mathrm{2}} −\mathrm{4}{x}\right)^{\mathrm{2}} }\: \\ $$$$\:\:\left.\begin{matrix}{{local}\:{maximum}}\\{{absolut}\:{maximum}}\end{matrix}\right\}\:=? \\ $$

Question Number 163099    Answers: 1   Comments: 0

1. u_(x x) = x + y 2. u_(x y) = x - y

$$\mathrm{1}.\:\mathrm{u}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{x}}} \:=\:\boldsymbol{\mathrm{x}}\:+\:\boldsymbol{\mathrm{y}} \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{u}}_{\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{y}}} \:=\:\boldsymbol{\mathrm{x}}\:-\:\boldsymbol{\mathrm{y}} \\ $$

Question Number 163101    Answers: 0   Comments: 0

Question Number 163100    Answers: 1   Comments: 0

Question Number 163075    Answers: 2   Comments: 0

((x^3 +1)/(x^2 −1)) = x+(√(6/x))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{x}^{\mathrm{3}} +\mathrm{1}}{{x}^{\mathrm{2}} −\mathrm{1}}\:=\:{x}+\sqrt{\frac{\mathrm{6}}{{x}}} \\ $$

Question Number 163061    Answers: 1   Comments: 0

(2)^(1/(log _x (((243)/x)))) = ((log _x ((x^5 /9))))^(1/3)

$$\:\sqrt[{\mathrm{log}\:_{{x}} \left(\frac{\mathrm{243}}{{x}}\right)}]{\mathrm{2}}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{log}\:_{{x}} \left(\frac{{x}^{\mathrm{5}} }{\mathrm{9}}\right)}\: \\ $$

Question Number 163060    Answers: 1   Comments: 1

2log _3 ((x^2 /(27))) = 2+ ((log _3 ((1/x)))/(log _5 ((√x) )))

$$\:\:\mathrm{2log}\:_{\mathrm{3}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{27}}\right)\:=\:\mathrm{2}+\:\frac{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{1}}{{x}}\right)}{\mathrm{log}\:_{\mathrm{5}} \left(\sqrt{{x}}\:\right)}\: \\ $$

Question Number 163053    Answers: 1   Comments: 1

Question Number 163048    Answers: 3   Comments: 0

Question Number 163045    Answers: 1   Comments: 0

prove that Σ_(n=1) ^∞ (( ( 2n+1 )!!)/((2n )!!)) (1/(2^( n) (2n +1)^( 2) )) =((π(√2))/4)−1

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(\:\mathrm{2}{n}+\mathrm{1}\:\right)!!}{\left(\mathrm{2}{n}\:\right)!!}\:\frac{\mathrm{1}}{\mathrm{2}^{\:{n}} \left(\mathrm{2}{n}\:+\mathrm{1}\right)^{\:\mathrm{2}} }\:=\frac{\pi\sqrt{\mathrm{2}}}{\mathrm{4}}−\mathrm{1} \\ $$

Question Number 163043    Answers: 0   Comments: 0

lim_(x→−∞) Σ_(n=1) ^∞ (x^n /n^n )=?

$$\underset{\mathrm{x}\rightarrow−\infty} {\mathrm{lim}}\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}^{\mathrm{n}} }=? \\ $$

Question Number 163042    Answers: 0   Comments: 0

lim_(n→∞) (√n)∫_(−∞) ^(+∞) ((cos x)/((1+x^2 )^n ))dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\int_{−\infty} ^{+\infty} \frac{\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }\mathrm{dx}=? \\ $$

Question Number 163041    Answers: 0   Comments: 0

lim_(n→∞) (√n)(∫_0 ^1 ((1−(1−x^2 )^n )/( (√n)x^2 ))dx−(√π))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }{\:\sqrt{\mathrm{n}}\mathrm{x}^{\mathrm{2}} }\mathrm{dx}−\sqrt{\pi}\right)=? \\ $$

Question Number 163040    Answers: 0   Comments: 0

lim_(n→∞) ((n!)/n^n )(Σ_(k=0) ^n (n^k /(k!))−Σ_(k=n+1) ^∞ (n^k /(k!)))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{n}!}{\mathrm{n}^{\mathrm{n}} }\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}^{\mathrm{k}} }{\mathrm{k}!}−\underset{\mathrm{k}=\mathrm{n}+\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{n}^{\mathrm{k}} }{\mathrm{k}!}\right)=? \\ $$

Question Number 163039    Answers: 0   Comments: 0

lim_(n→∞) (√n)∫_0 ^1 x∙sin^(2n) (2πx)dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{\mathrm{n}}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}\centerdot\mathrm{sin}^{\mathrm{2n}} \left(\mathrm{2}\pi\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 163072    Answers: 1   Comments: 0

∫((sinx+sin3x+sin5x+sin7x)/(cosx+cos3x+cos5x+cos7x)) dx

$$\int\frac{\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{sin}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{7}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{cosx}}+\boldsymbol{\mathrm{cos}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{7}\boldsymbol{\mathrm{x}}}\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 163033    Answers: 1   Comments: 0

Ω= ∫_0 ^( ∞) (( x − sin (x ))/x^( 3) )dx −−− solution−−− Ω=^(I.B.P) [ ((−1)/(2 x^( 2) )) (x−sin(x))]_0 ^∞ +(1/2) ∫_0 ^( ∞) ((1−cos (x))/x^( 2) )dx = (1/2) ∫_0 ^( ∞) (( 2sin^( 2) ((x/2)))/x^( 2) )dx=∫_0 ^( ∞) ((sin^( 2) ((x/2)))/x^( 2) )dx =^((x/2) = α) (1/2)∫_0 ^( ∞) ((sin^( 2) ( α))/α^( 2) ) dα = (1/2) [((−1)/α) sin^( 2) (α)]_0 ^∞ +(1/2)∫_0 ^( ∞) ((sin(2α))/α)dα =^(2α=ϕ) (1/2) ∫_0 ^( ∞) (( sin(ϕ ))/ϕ) dϕ =(π/4) −− Ω= (π/4) −−−

$$ \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}\:−\:{sin}\:\left({x}\:\right)}{{x}^{\:\mathrm{3}} }{dx} \\ $$$$−−−\:{solution}−−− \\ $$$$\:\:\:\:\:\Omega\overset{\mathscr{I}.\mathscr{B}.\mathscr{P}} {=}\:\left[\:\frac{−\mathrm{1}}{\mathrm{2}\:{x}^{\:\mathrm{2}} }\:\left({x}−{sin}\left({x}\right)\right)\right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}−{cos}\:\left({x}\right)}{{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{2}{sin}^{\:\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{{x}^{\:\mathrm{2}} }{dx}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\overset{\frac{{x}}{\mathrm{2}}\:=\:\alpha} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{2}} \left(\:\alpha\right)}{\alpha^{\:\mathrm{2}} }\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\left[\frac{−\mathrm{1}}{\alpha}\:{sin}^{\:\mathrm{2}} \left(\alpha\right)\right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}\alpha\right)}{\alpha}{d}\alpha \\ $$$$\:\:\:\:\:\:\:\:\overset{\mathrm{2}\alpha=\varphi} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left(\varphi\:\right)}{\varphi}\:{d}\varphi\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−\:\:\:\:\:\Omega=\:\frac{\pi}{\mathrm{4}}\:\:−−− \\ $$$$ \\ $$$$ \\ $$

Question Number 163024    Answers: 0   Comments: 2

calcul en fonction de n Σ_(k=0) ^(2n) (_k ^(2n) )^(n−2k)

$${calcul}\:{en}\:{fonction}\:{de}\:{n} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}\left(_{{k}} ^{\mathrm{2}{n}} \right)^{{n}−\mathrm{2}{k}} \\ $$

Question Number 163020    Answers: 0   Comments: 2

cos^(2n) (x)+sin^(2n) (x)=((4^n +1)/5^n ) prove that

$$\boldsymbol{\mathrm{cos}}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)+\boldsymbol{\mathrm{sin}}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{4}^{\boldsymbol{\mathrm{n}}} +\mathrm{1}}{\mathrm{5}^{\boldsymbol{\mathrm{n}}} } \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$

Question Number 163014    Answers: 2   Comments: 0

given (a/b)=(c/d), find an expression for (a/(a+b))

$${given}\:\frac{{a}}{{b}}=\frac{{c}}{{d}},\:{find}\:{an}\:{expression} \\ $$$${for}\:\:\frac{{a}}{{a}+{b}} \\ $$

Question Number 163028    Answers: 1   Comments: 0

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