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Question Number 37349 Answers: 0 Comments: 1

calculate ∫_0 ^(2π) (dt/(1−2pcost +p^2 )) if ∣p∣<1

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\mathrm{1}−\mathrm{2}{pcost}\:+{p}^{\mathrm{2}} }\:\:{if}\:\mid{p}\mid<\mathrm{1} \\ $$

Question Number 37348 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(p +cost)) with p>1

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dt}}{{p}\:+{cost}}\:\:{with}\:{p}>\mathrm{1} \\ $$

Question Number 37347 Answers: 1 Comments: 2

let r =(√(p^2 +q^2 )) p and q from R and p>0 q>0 1)prove that ∫_0 ^(+∞) e^(−px) ((cos(px))/(√x))dx=((√π)/r)(√((r+p)/2)) 2) ∫_0 ^∞ e^(−px) ((sin(qx))/(√x))dx =((√π)/r) (√((r−p)/2))

$${let}\:{r}\:=\sqrt{{p}^{\mathrm{2}} \:+{q}^{\mathrm{2}} }\:\:\:{p}\:{and}\:{q}\:{from}\:{R}\:\:{and}\:{p}>\mathrm{0}\:\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:{e}^{−{px}} \:\frac{{cos}\left({px}\right)}{\sqrt{{x}}}{dx}=\frac{\sqrt{\pi}}{{r}}\sqrt{\frac{{r}+{p}}{\mathrm{2}}} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\frac{{sin}\left({qx}\right)}{\sqrt{{x}}}{dx}\:=\frac{\sqrt{\pi}}{{r}}\:\sqrt{\frac{{r}−{p}}{\mathrm{2}}} \\ $$

Question Number 37346 Answers: 0 Comments: 0

find the value of ∫_0 ^(2π) (dt/(a cos^2 t +b sin^2 t)) with a>0 and b>0 .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+{b}\:{sin}^{\mathrm{2}} {t}} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:. \\ $$

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 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 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 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)}

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

Question Number 37297 Answers: 0 Comments: 1

calculate ∫_C (z/(z^2 +1))dz with C={z∈C/∣z∣=(1/2)}

$${calculate}\:\:\int_{{C}} \:\:\:\:\frac{{z}}{{z}^{\mathrm{2}} \:+\mathrm{1}}{dz}\:\:{with}\:{C}=\left\{{z}\in{C}/\mid{z}\mid=\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$

Question Number 37291 Answers: 0 Comments: 1

calculate g(θ) = ∫_(−∞) ^(+∞) e^(−x^2 ) sin(sinθ x^2 )dx .

$${calculate}\:{g}\left(\theta\right)\:=\:\int_{−\infty} ^{+\infty} \:{e}^{−{x}^{\mathrm{2}} } \:{sin}\left({sin}\theta\:{x}^{\mathrm{2}} \right){dx}\:. \\ $$

Question Number 37290 Answers: 0 Comments: 0

find f(θ) = ∫_(−∞) ^(+∞) e^(−x^2 ) cos(cosθx)dx .

$${find}\:\:{f}\left(\theta\right)\:=\:\int_{−\infty} ^{+\infty} \:{e}^{−{x}^{\mathrm{2}} } \:{cos}\left({cos}\theta{x}\right){dx}\:. \\ $$

Question Number 37289 Answers: 0 Comments: 0

calculate ∫_0 ^(2π) (dx/(cos^2 t +4sin^2 t))dt .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\:\frac{{dx}}{{cos}^{\mathrm{2}} {t}\:\:+\mathrm{4}{sin}^{\mathrm{2}} {t}}{dt}\:. \\ $$

Question Number 37288 Answers: 0 Comments: 1

calculate f(α) = ∫_(−∞) ^(+∞) ((cos(2x))/(1+ax^2 )) dx with a>0 2) find the value of ∫_(−∞) ^(+∞) ((cos(2x))/(1+3x^2 )) dx .

$${calculate}\:\:{f}\left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{ax}^{\mathrm{2}} }\:{dx}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

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