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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 36191    Answers: 0   Comments: 1

let D = {(x,y)∈R^2 /x>0 ,y>0,x+y<1} 1) calculate ∫∫_D ((xy)/(x^2 +y^2 ))dxdy 2) let a>0 ,b>0 calculate ∫∫_D a^x b^y dxdy

$${let}\:{D}\:=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/{x}>\mathrm{0}\:,{y}>\mathrm{0},{x}+{y}<\mathrm{1}\right\} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int\int_{{D}} \:\:\frac{{xy}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{a}>\mathrm{0}\:,{b}>\mathrm{0}\:{calculate}\:\int\int_{{D}} \:{a}^{{x}} {b}^{{y}} {dxdy} \\ $$

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 36179    Answers: 0   Comments: 1

let f(x,y) = ((xy)/(x+y)) 1) find D_f 2)calcule x(∂f/∂x)(x,y) +y (∂f/∂y)(x,y) interms of f(x,y)

$${let}\:{f}\left({x},{y}\right)\:=\:\frac{{xy}}{{x}+{y}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right){calcule}\:{x}\frac{\partial{f}}{\partial{x}}\left({x},{y}\right)\:+{y}\:\frac{\partial{f}}{\partial{y}}\left({x},{y}\right)\:{interms}\:{of}\:{f}\left({x},{y}\right) \\ $$

Question Number 36178    Answers: 0   Comments: 1

let f(x,y)=ln((√(x^2 +y^2 ))) calculate (∂^2 f/∂x^2 )(x,y)+(∂^2 f/∂y^2 )

$${let}\:{f}\left({x},{y}\right)={ln}\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\right)\: \\ $$$${calculate}\:\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }\left({x},{y}\right)+\frac{\partial^{\mathrm{2}} {f}}{\partial{y}^{\mathrm{2}} } \\ $$

Question Number 36177    Answers: 0   Comments: 1

let f(x)= arctan((x/y)) calculate (∂^2 f/∂x^2 )(x,y) , (∂^2 f/∂y^2 )(x,y), (∂^2 f/(∂x∂y))(x,y) (∂^2 f/(∂y∂x))(x,y)

$${let}\:{f}\left({x}\right)=\:{arctan}\left(\frac{{x}}{{y}}\right) \\ $$$${calculate}\:\:\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }\left({x},{y}\right)\:,\:\frac{\partial^{\mathrm{2}} {f}}{\partial{y}^{\mathrm{2}} }\left({x},{y}\right),\:\frac{\partial^{\mathrm{2}} {f}}{\partial{x}\partial{y}}\left({x},{y}\right) \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{y}\partial{x}}\left({x},{y}\right) \\ $$

Question Number 36176    Answers: 0   Comments: 0

let f(x,y) =(x^2 +y^2 )sin{ (1/(√(x^2 +y^2 )))} if(x,y)=(0,0) and f(0,0)=0 prove that f is differenciable at all point of R^2 2) prove that (∂f/∂x) and (∂f/∂y) are not differdnciable at (0,0)

$${let}\:{f}\left({x},{y}\right)\:=\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){sin}\left\{\:\frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\right\}\:{if}\left({x},{y}\right)=\left(\mathrm{0},\mathrm{0}\right) \\ $$$${and}\:{f}\left(\mathrm{0},\mathrm{0}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:{f}\:{is}\:{differenciable}\:{at}\:{all}\:{point}\:{of}\:{R}^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\partial{f}}{\partial{x}}\:{and}\:\frac{\partial{f}}{\partial{y}}\:{are}\:{not}\:{differdnciable} \\ $$$${at}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$

Question Number 36175    Answers: 0   Comments: 0

let g(x,y) = ((1+x+y)/(x^2 −y^2 )) is g have a limit at (0,0)?

$${let}\:{g}\left({x},{y}\right)\:=\:\frac{\mathrm{1}+{x}+{y}}{{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} } \\ $$$${is}\:{g}\:{have}\:{a}\:{limit}\:{at}\:\left(\mathrm{0},\mathrm{0}\right)? \\ $$

Question Number 36174    Answers: 0   Comments: 0

find lim_((x,y)→(0,0)) ((1−cos((√(xy))))/y)

$${find}\:{lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\:\frac{\mathrm{1}−{cos}\left(\sqrt{{xy}}\right)}{{y}} \\ $$

Question Number 36173    Answers: 0   Comments: 1

calculate (∂f/∂x) and (∂f/∂y) in this cases 1) f(x,y)= e^(−x) sin(2y +1) 2)f(x,y) =(x^2 +y^2 )e^(−xy) 3)f(x,y) = (x/(x^2 +y^2 ))

$${calculate}\:\frac{\partial{f}}{\partial{x}}\:{and}\:\frac{\partial{f}}{\partial{y}}\:{in}\:{this}\:{cases} \\ $$$$\left.\mathrm{1}\right)\:{f}\left({x},{y}\right)=\:{e}^{−{x}} \:{sin}\left(\mathrm{2}{y}\:+\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){f}\left({x},{y}\right)\:=\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){e}^{−{xy}} \\ $$$$\left.\mathrm{3}\right){f}\left({x},{y}\right)\:=\:\frac{{x}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} } \\ $$

Question Number 36168    Answers: 0   Comments: 3

let A(t) = ∫_(−∞) ^(+∞) ((sin(xt))/(( x +1+i)^2 )) dx with t from R 2) calculate A(t) 2) extract Re(A(t)) and Im(A(t)) 3) find the value of ∫_(−∞) ^(+∞) ((cos(3x))/((x+1+i)^2 ))dx

$${let}\:{A}\left({t}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{sin}\left({xt}\right)}{\left(\:{x}\:+\mathrm{1}+{i}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:{t}\:{from}\:{R} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{extract}\:{Re}\left({A}\left({t}\right)\right)\:{and}\:{Im}\left({A}\left({t}\right)\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{cos}\left(\mathrm{3}{x}\right)}{\left({x}+\mathrm{1}+{i}\right)^{\mathrm{2}} }{dx} \\ $$

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