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Question Number 37345 Answers: 0 Comments: 0

calculate I(a) = ∫_0 ^(2π) ((1+acost)/(1+2acost +a^2 ))dt 1) if ∣a∣<1 2) if ∣a∣>1

$${calculate}\:{I}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{\mathrm{1}+{acost}}{\mathrm{1}+\mathrm{2}{acost}\:+{a}^{\mathrm{2}} }{dt}\:\: \\ $$$$\left.\mathrm{1}\right)\:{if}\:\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{if}\:\mid{a}\mid>\mathrm{1} \\ $$

Question Number 37344 Answers: 0 Comments: 1

solve the d.e. y^′ −xy =cosx .

$${solve}\:{the}\:{d}.{e}.\:{y}^{'} \:−{xy}\:\:={cosx}\:. \\ $$

Question Number 37343 Answers: 0 Comments: 3

let f(x) = ∫_0 ^1 ln(1+xt^2 )dt with ∣x∣<1 1) find f(x) 2) calculate ∫_0 ^1 ln(2+t^2 )dt .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 37342 Answers: 0 Comments: 2

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1))) x^n with ∣x∣<1 2) find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)2^n )) .

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:{x}^{{n}} \:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 37341 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (3/(n^2 (2n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{3}}{{n}^{\mathrm{2}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 37339 Answers: 0 Comments: 1

find the value of Σ_(n=1) ^∞ ((2n+1)/(1 +2^3 +3^3 +...+n^3 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\:\:\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{1}\:+\mathrm{2}^{\mathrm{3}} \:+\mathrm{3}^{\mathrm{3}} \:+...+{n}^{\mathrm{3}} } \\ $$

Question Number 37338 Answers: 0 Comments: 1

calculate B_n = ∫_0 ^1 sh^n xdx .

$${calculate}\:\:{B}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{sh}^{{n}} {xdx}\:. \\ $$

Question Number 37337 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ch^n xdx .

$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ch}^{{n}} {xdx}\:. \\ $$

Question Number 37335 Answers: 0 Comments: 1

find ∫ x arctan(x+(1/x))dx .

$${find}\:\int\:\:\:\:\:{x}\:{arctan}\left({x}+\frac{\mathrm{1}}{{x}}\right){dx}\:. \\ $$

Question Number 37334 Answers: 0 Comments: 1

study the convergence of Σ_(n=0) ^∞ e^(−x) Σ_(k=0) ^∞ (x^k /(k!)) .

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \:\sum_{{k}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$

Question Number 37333 Answers: 0 Comments: 2

let f(x)=Σ_(n=1) ^∞ ((sin(nx))/n^3 ) 1)study the convergence of this serie 2)prove that ∫_0 ^π f(x)dx=2 Σ_(n=1) ^∞ (1/((2n−1)^4 )) 3)prove that ∀x∈ ∈R f^′ (x)=Σ_(n=1) ^∞ ((cos(nx))/n^2 ) 4) prove that ∫_0 ^(π/2) ( Σ_(n≥1) ((cos(nx))/n^2 ))=Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^2 ))

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} } \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall{x}\in\:\in{R}\:\:{f}^{'} \left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\:\sum_{{n}\geqslant\mathrm{1}} \frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 37324 Answers: 0 Comments: 3

Question Number 46453 Answers: 1 Comments: 6

Question Number 37317 Answers: 2 Comments: 4

∫ ((acos x+b)/((a+bcos x)^2 ))dx = ?

$$\int\:\frac{\mathrm{acos}\:{x}+{b}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}\:=\:? \\ $$

Question Number 37316 Answers: 1 Comments: 0

∫ (x^2 /((xsin x+cos x)^2 ))dx = ?

$$\int\:\frac{{x}^{\mathrm{2}} }{\left({x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}\:=\:? \\ $$

Question Number 37310 Answers: 1 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +1)(x^2 +4)(x^2 +9))) .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)\left({x}^{\mathrm{2}} \:+\mathrm{9}\right)}\:. \\ $$

Question Number 37309 Answers: 1 Comments: 2

calculate ∫_0 ^(+∞) (x^2 /((1+x^2 )^3 )) dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:{dx}\:. \\ $$

Question Number 37307 Answers: 0 Comments: 1

calculate ∫_γ (dz/z) with γ ={z∈C /∣z∣=1} .

$${calculate}\:\:\:\int_{\gamma} \:\:\:\:\frac{{dz}}{{z}}\:\:\:{with}\:\gamma\:=\left\{{z}\in{C}\:/\mid{z}\mid=\mathrm{1}\right\}\:. \\ $$

Question Number 37306 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) e^(ix) ((x−i)/((x+i)(x^2 +3))) dx .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:{e}^{{ix}} \:\:\:\frac{{x}−{i}}{\left({x}+{i}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)}\:{dx}\:. \\ $$$$ \\ $$

Question Number 37304 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((5+e^(ix) )/((3+e^(ix) )(1+x^2 )))dx .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{5}+{e}^{{ix}} }{\left(\mathrm{3}+{e}^{{ix}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 37303 Answers: 0 Comments: 1

calculate ∫_γ (dz/(z^3 +8)) in those cases 1) γ ={z∈C / ∣z∣ =1} 2) γ ={z∈C / ∣z∣ =3}

$${calculate}\:\:\int_{\gamma} \:\:\:\:\:\:\frac{{dz}}{{z}^{\mathrm{3}} \:+\mathrm{8}}\:{in}\:{those}\:{cases} \\ $$$$\left.\mathrm{1}\right)\:\gamma\:=\left\{{z}\in{C}\:/\:\mid{z}\mid\:=\mathrm{1}\right\} \\ $$$$\left.\mathrm{2}\right)\:\gamma\:=\left\{{z}\in{C}\:/\:\mid{z}\mid\:=\mathrm{3}\right\} \\ $$

Question Number 37302 Answers: 0 Comments: 0

let γ = {z∈C / ∣z∣ =4} calculate ∫_γ (dz/(z sinz)) in the positif sens.

$${let}\:\gamma\:=\:\left\{{z}\in{C}\:/\:\mid{z}\mid\:=\mathrm{4}\right\}\: \\ $$$${calculate}\:\:\int_{\gamma} \:\:\:\:\:\frac{{dz}}{{z}\:{sinz}}\:{in}\:{the}\:{positif}\:{sens}. \\ $$

Question Number 37301 Answers: 0 Comments: 0

find?the value of ∫_(−∞) ^(+∞) (((2x+1)e^(−x^2 ) )/(1+4x^2 )) dx .

$${find}?{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\left(\mathrm{2}{x}+\mathrm{1}\right){e}^{−{x}^{\mathrm{2}} } }{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 37300 Answers: 0 Comments: 1

let f(z)=(((1−z^2 )e^(2z) )/z^3 ) calculate Res(f, 0)

$${let}\:{f}\left({z}\right)=\frac{\left(\mathrm{1}−{z}^{\mathrm{2}} \right){e}^{\mathrm{2}{z}} }{{z}^{\mathrm{3}} } \\ $$$${calculate}\:{Res}\left({f},\:\mathrm{0}\right) \\ $$

Question Number 37299 Answers: 0 Comments: 1

calculate ∫_C ((9(z^2 +2))/(z(z+1)^3 (z−2)))dz with C is the circle C ={z∈C/ ∣z∣ =3}

$${calculate}\:\:\int_{{C}} \:\:\:\frac{\mathrm{9}\left({z}^{\mathrm{2}} \:+\mathrm{2}\right)}{{z}\left({z}+\mathrm{1}\right)^{\mathrm{3}} \left({z}−\mathrm{2}\right)}{dz}\:\:{with}\:\:{C}\:{is}\:{the} \\ $$$${circle}\:{C}\:=\left\{{z}\in{C}/\:\mid{z}\mid\:=\mathrm{3}\right\}\: \\ $$

Question Number 37298 Answers: 0 Comments: 1

calculate ∫_γ ((z+1)/(z(z−1)(z+2)))dz with γ is the circle γ ={z∈C/ ∣z∣ =(3/2)}

$${calculate}\:\:\int_{\gamma} \:\:\:\:\frac{{z}+\mathrm{1}}{{z}\left({z}−\mathrm{1}\right)\left({z}+\mathrm{2}\right)}{dz}\:\:{with}\:\gamma\:{is}\:{the} \\ $$$${circle}\:\gamma\:=\left\{{z}\in{C}/\:\:\mid{z}\mid\:=\frac{\mathrm{3}}{\mathrm{2}}\right\} \\ $$

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