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Question Number 65457    Answers: 1   Comments: 0

{ (((a/b)+((b−a)/(b+a))=2(√3))),(((b/a)+((b+a)/(b−a))=3(√2))) :} [a,b∈R,a≠b]

$$\begin{cases}{\frac{\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{b}}}+\frac{\boldsymbol{\mathrm{b}}−\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}=\mathrm{2}\sqrt{\mathrm{3}}}\\{\frac{\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{a}}}+\frac{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{b}}−\boldsymbol{\mathrm{a}}}=\mathrm{3}\sqrt{\mathrm{2}}}\end{cases}\:\:\:\:\:\left[\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\boldsymbol{\mathrm{R}},\boldsymbol{\mathrm{a}}\neq\boldsymbol{\mathrm{b}}\right] \\ $$

Question Number 65455    Answers: 0   Comments: 2

find U_n = ∫_0 ^(+∞) (((−1)^x )/(2^x^2 (x^2 +4n^2 )))dx (n from N and n≥1) study nature of the serie Σ 2^n^2 U_n

$${find}\:{U}_{{n}} =\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{x}} }{\mathrm{2}^{{x}^{\mathrm{2}} } \left({x}^{\mathrm{2}} \:+\mathrm{4}{n}^{\mathrm{2}} \right)}{dx}\:\:\:\:\:\left({n}\:{from}\:{N}\:{and}\:{n}\geqslant\mathrm{1}\right) \\ $$$${study}\:{nature}\:{of}\:{the}\:{serie}\:\:\Sigma\:\mathrm{2}^{{n}^{\mathrm{2}} } {U}_{{n}} \\ $$

Question Number 65450    Answers: 0   Comments: 1

Question Number 65446    Answers: 0   Comments: 1

10^x = x^(1000) ⇒ x=?

$$\:\:\:\:\mathrm{10}^{\mathrm{x}} \:=\:\mathrm{x}^{\mathrm{1000}} \:\Rightarrow\:\mathrm{x}=? \\ $$

Question Number 65445    Answers: 0   Comments: 2

find f(x) =∫_0 ^(π/4) ln(cost +xsint)dt

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({cost}\:+{xsint}\right){dt}\:\:\: \\ $$

Question Number 65443    Answers: 0   Comments: 0

Question Number 65427    Answers: 0   Comments: 1

Question Number 65423    Answers: 1   Comments: 1

Question Number 65420    Answers: 0   Comments: 0

Question Number 65414    Answers: 0   Comments: 1

Question Number 65403    Answers: 0   Comments: 0

solve the(de) x^3 y^(′′) −2xy^′ +(x+1)y =0

$${solve}\:{the}\left({de}\right)\:\:\:\:\:\:{x}^{\mathrm{3}} {y}^{''} −\mathrm{2}{xy}^{'} \:+\left({x}+\mathrm{1}\right){y}\:=\mathrm{0} \\ $$

Question Number 65402    Answers: 0   Comments: 1

solve the (de) (√(x^2 +1))y^(′′) −xy^′ =x^2 −x

$${solve}\:{the}\:\left({de}\right)\:\:\:\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}{y}^{''} \:\:\:−{xy}^{'} \:={x}^{\mathrm{2}} −{x} \\ $$

Question Number 65401    Answers: 0   Comments: 1

let f(x,y)=(x+y)(√(x+y−1)) calculate ∫∫_D f(x,y)dxdy with D ={(x,y)∈R^2 / 1≤x≤2 and 1≤y≤(√3)}

$${let}\:{f}\left({x},{y}\right)=\left({x}+{y}\right)\sqrt{{x}+{y}−\mathrm{1}} \\ $$$${calculate}\:\:\int\int_{{D}} {f}\left({x},{y}\right){dxdy}\:{with}\: \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:\mathrm{1}\leqslant{x}\leqslant\mathrm{2}\:\:{and}\:\:\:\mathrm{1}\leqslant{y}\leqslant\sqrt{\mathrm{3}}\right\} \\ $$

Question Number 65400    Answers: 0   Comments: 0

find f(α) =∫_1 ^(+∞) ((arctan((α/x)))/(1+x^2 )) dx with α≥0

$${find}\:{f}\left(\alpha\right)\:=\int_{\mathrm{1}} ^{+\infty} \:\frac{{arctan}\left(\frac{\alpha}{{x}}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:\:\:{with}\:\alpha\geqslant\mathrm{0} \\ $$

Question Number 65398    Answers: 0   Comments: 1

1) calculate A_n =∫∫_([1,n[^2 ) sin(x^2 +3y^2 ) e^(−x^2 −3y^2 ) dxdy 2) determine lim_(n→+∞) A_n

$$\left.\mathrm{1}\right)\:{calculate}\:\:{A}_{{n}} =\int\int_{\left[\mathrm{1},{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:\:{sin}\left({x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} \right)\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } {dxdy} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 65399    Answers: 0   Comments: 1

1) calculate A_n = ∫∫_([0,n[^2 ) ((dxdy)/(√(x^2 +y^2 +4))) 2)find lim_(n→+∞) A_n

$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} =\:\int\int_{\left[\mathrm{0},{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\frac{{dxdy}}{\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 65407    Answers: 0   Comments: 1

Question Number 65405    Answers: 0   Comments: 1

Question Number 65395    Answers: 1   Comments: 1

Question Number 65383    Answers: 0   Comments: 1

1) calculate ∫_0 ^∞ (dx/(1+e^(nx) )) with n integr natural and n≥1 2) conclude the value of Σ_(k=1) ^∞ (((−1)^(k−1) )/k)

$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{e}^{{nx}} }\:\:\:{with}\:{n}\:{integr}\:{natural}\:\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}} \\ $$

Question Number 65380    Answers: 1   Comments: 1

Question Number 65387    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((arctan(2x)−arctanx)/x)dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{x}\right)−{arctanx}}{{x}}{dx} \\ $$

Question Number 65386    Answers: 0   Comments: 1

calculate Σ_(k=1) ^∞ (((−1)^(k−1) )/(99k−1))

$${calculate}\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{\mathrm{99}{k}−\mathrm{1}} \\ $$

Question Number 65372    Answers: 0   Comments: 4

I=∫_1 ^e (dx/(x(1+ln^2 x)))

$${I}=\int_{\mathrm{1}} ^{{e}} \:\frac{{dx}}{{x}\left(\mathrm{1}+{ln}^{\mathrm{2}} {x}\right)} \\ $$

Question Number 65367    Answers: 2   Comments: 1

Question Number 65366    Answers: 2   Comments: 2

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