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IntegrationQuestion and Answers: Page 295

Question Number 36202    Answers: 0   Comments: 1

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

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

Question Number 36201    Answers: 0   Comments: 1

calculate ∫_0 ^(π/2) (dθ/(1+2sin^2 θ))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{d}\theta}{\mathrm{1}+\mathrm{2}{sin}^{\mathrm{2}} \theta} \\ $$

Question Number 36200    Answers: 0   Comments: 4

calculate ∫_0 ^(2π) (dθ/((2+cosθ)^2 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{d}\theta}{\left(\mathrm{2}+{cos}\theta\right)^{\mathrm{2}} } \\ $$

Question Number 36198    Answers: 0   Comments: 1

let f(z) = ((z^2 +1)/(z^4 −1)) find (a_(k)) the poles of f and calculate Res(f,a_k )

$${let}\:{f}\left({z}\right)\:=\:\frac{{z}^{\mathrm{2}} \:+\mathrm{1}}{{z}^{\mathrm{4}} −\mathrm{1}} \\ $$$${find}\:\left({a}_{\left.{k}\right)} {the}\:{poles}\:{of}\:{f}\:{and}\:{calculate}\:\right. \\ $$$${Res}\left({f},{a}_{{k}} \right) \\ $$

Question Number 36197    Answers: 0   Comments: 1

find the value of ∫_0 ^(2π) (dx/(cos^2 x +3 sin^2 x))

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dx}}{{cos}^{\mathrm{2}} {x}\:+\mathrm{3}\:{sin}^{\mathrm{2}} {x}} \\ $$

Question Number 36196    Answers: 0   Comments: 0

let ρ>0 and C the circle x^2 +y^2 =ρ^2 calculate ∫_C ydx +xy dy

$${let}\:\rho>\mathrm{0}\:\:{and}\:{C}\:{the}\:{circle}\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:=\rho^{\mathrm{2}} \\ $$$${calculate}\:\int_{{C}} \:{ydx}\:+{xy}\:{dy} \\ $$

Question Number 36195    Answers: 0   Comments: 1

let C ={(x,y)∈R^2 / 0≤x≤1 and y=2x^2 } calculate ∫_C x^2 ydx +(x^2 −y^2 )dy

$${let}\:\:{C}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:{y}=\mathrm{2}{x}^{\mathrm{2}} \right\} \\ $$$${calculate}\:\int_{{C}} \:{x}^{\mathrm{2}} {ydx}\:+\left({x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} \right){dy} \\ $$

Question Number 36194    Answers: 0   Comments: 0

let D ={(x,y,z)∈R^2 / 0<z<1 and x^2 +y^2 <z^2 } calculate ∫∫_D xyzdxdydz

$${let}\:{D}\:=\left\{\left({x},{y},{z}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}<{z}<\mathrm{1}\:{and}\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:<{z}^{\mathrm{2}} \right\} \\ $$$${calculate}\:\int\int_{{D}} {xyzdxdydz} \\ $$

Question Number 36193    Answers: 0   Comments: 1

let D ={(x,y)∈ R^2 / x^2 +y^2 −x<0 and x^2 +y^2 −y >0 and y>0} calculate∫∫_D (x+y)^2 dxdy

$${let}\:{D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:−{x}<\mathrm{0}\:{and}\right. \\ $$$$\left.{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:−{y}\:>\mathrm{0}\:{and}\:{y}>\mathrm{0}\right\} \\ $$$${calculate}\int\int_{{D}} \:\:\left({x}+{y}\right)^{\mathrm{2}} {dxdy} \\ $$

Question Number 36192    Answers: 0   Comments: 1

let D ={(x,y)∈ R^2 /x^2 +y^2 <1} find the value of ∫∫_D ((dxdy)/(x^2 +y^(2 ) + 2))

$${let}\:{D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} <\mathrm{1}\right\} \\ $$$${find}\:{the}\:{value}\:{of}\:\int\int_{{D}} \:\frac{{dxdy}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}\:} +\:\mathrm{2}} \\ $$

Question Number 36190    Answers: 0   Comments: 1

calculate ∫∫_D (x+y)e^(x+y) dxdy with D = {(x,y)∈R^2 / 0<x<2 and 1<y<2 }

$${calculate}\:\:\int\int_{{D}} \left({x}+{y}\right){e}^{{x}+{y}} {dxdy}\:\:{with} \\ $$$${D}\:=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\mathrm{0}<{x}<\mathrm{2}\:{and}\:\:\mathrm{1}<{y}<\mathrm{2}\:\right\} \\ $$

Question Number 36189    Answers: 0   Comments: 1

let F(x)=∫_0 ^∞ ((e^(−x^2 t) (√t))/(1+t^2 ))dt calculate lim_(x→+∞) F(x) .

$${let}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{x}^{\mathrm{2}} {t}} \sqrt{{t}}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$${calculate}\:{lim}_{{x}\rightarrow+\infty} \:{F}\left({x}\right)\:. \\ $$$$ \\ $$

Question Number 36188    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ ((√t)/(1+t^2 ))dt

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\sqrt{{t}}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 36187    Answers: 0   Comments: 3

let I_n (x)= ∫_0 ^∞ ((t sin(t))/((t^2 +x^2 )^n ))dt 1) find a relation between I_(n+1) and I_n 2) calculate I_2 (x) and I_3 (x) 3) calculate ∫_0 ^∞ ((tsin(t))/((2+t^2 )^2 ))dt

$${let}\:{I}_{{n}} \left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}\:{sin}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{{n}} }{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{relation}\:{between}\:{I}_{{n}+\mathrm{1}} \:\:{and}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{\mathrm{2}} \left({x}\right)\:{and}\:{I}_{\mathrm{3}} \left({x}\right)\: \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{tsin}\left({t}\right)}{\left(\mathrm{2}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$

Question Number 36186    Answers: 0   Comments: 1

find nature of ∫_1 ^(+∞) (√t) sin(t^2 )dt .

$${find}\:{nature}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \sqrt{{t}}\:{sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 36185    Answers: 0   Comments: 2

study the vonvergence of ∫_1 ^(+∞) ((e^(−(1/t)) −cos((1/t)))/t)dt

$${study}\:{the}\:{vonvergence}\:{of}\: \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{−\frac{\mathrm{1}}{{t}}} \:−{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}{dt} \\ $$

Question Number 36184    Answers: 0   Comments: 1

study the convergence of ∫_1 ^(+∞) ((cos(t))/(√t))dt

$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{cos}\left({t}\right)}{\sqrt{{t}}}{dt} \\ $$

Question Number 36183    Answers: 0   Comments: 0

calculate ∫_1 ^(+∞) arctan((1/t))dt

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:{arctan}\left(\frac{\mathrm{1}}{{t}}\right){dt} \\ $$

Question Number 36182    Answers: 2   Comments: 1

calculate ∫_1 ^(+∞) (dt/(t(√(1+t^2 ))))

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$

Question Number 36181    Answers: 0   Comments: 1

let I(ξ) = ∫_ξ ^(1−ξ) (dt/(1−(t−ξ)^2 )) find lim_(ξ→0^+ ) I(ξ)

$${let}\:{I}\left(\xi\right)\:\:=\:\int_{\xi} ^{\mathrm{1}−\xi} \:\:\:\frac{{dt}}{\mathrm{1}−\left({t}−\xi\right)^{\mathrm{2}} } \\ $$$${find}\:{lim}_{\xi\rightarrow\mathrm{0}^{+} } \:\:\:{I}\left(\xi\right) \\ $$

Question Number 36180    Answers: 1   Comments: 1

calculate ∫_0 ^1 ((ln(t))/((1+t)^2 ))dt

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 36167    Answers: 0   Comments: 2

let give I = ∫_0 ^∞ (dx/((x^2 +i)^2 )) 1) extract Re(I) and Im(I) 2) find the value of I 3) calculate Re(I) and Im(I) .

$${let}\:{give}\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right)\:. \\ $$

Question Number 36057    Answers: 2   Comments: 1

find the value of ∫_0 ^(π/4) ((cosx)/(sinx +tanx))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{cosx}}{{sinx}\:+{tanx}}{dx}\: \\ $$

Question Number 36056    Answers: 1   Comments: 2

calculate ∫_(−∞) ^(+∞) ((2x)/((x^2 +mx +1)^2 ))dx with ∣m∣<2

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}}{\left({x}^{\mathrm{2}} \:+{mx}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{with}\:\mid{m}\mid<\mathrm{2} \\ $$

Question Number 36031    Answers: 0   Comments: 0

Q. Evaluate: ∫_(∫xyzdxdydz) ^(∫zyxdzdydx) ∫_((d/dx)(x^(sin x) )) ^((d/dx)(x^(cos x) )) ∫_(lim_(x→0) ((−x^2 +2)/x)) ^(lim_(x→0) ((x^2 −2)/x)) ∫_0 ^∞ w^(1−x) x^(1−y) y^(1−z) z^(1−w) dwdxdydz

$$\mathrm{Q}.\:\mathrm{Evaluate}:\:\:\:\int_{\int\mathrm{xyzdxdydz}} ^{\int\mathrm{zyxdzdydx}} \int_{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{sin}\:\mathrm{x}} \right)} ^{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{cos}\:\mathrm{x}} \right)} \int_{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}}} ^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}}} \int_{\mathrm{0}} ^{\infty} \mathrm{w}^{\mathrm{1}−\mathrm{x}} \mathrm{x}^{\mathrm{1}−\mathrm{y}} \mathrm{y}^{\mathrm{1}−\mathrm{z}} \mathrm{z}^{\mathrm{1}−\mathrm{w}} \mathrm{dwdxdydz} \\ $$

Question Number 36009    Answers: 0   Comments: 1

calculate ∫_(−∞) ^(+∞) ((xdx)/((2x+1+i)^3 )) with i^2 =−1 .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}+{i}\right)^{\mathrm{3}} }\:\:{with}\:{i}^{\mathrm{2}} \:=−\mathrm{1}\:. \\ $$

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