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IntegrationQuestion and Answers: Page 292

Question Number 36418 Answers: 1 Comments: 0

calculate ∫_(−3) ^4 ∣x^2 −2x−3∣dx

$${calculate}\:\:\int_{−\mathrm{3}} ^{\mathrm{4}} \mid{x}^{\mathrm{2}} \:−\mathrm{2}{x}−\mathrm{3}\mid{dx} \\ $$

Question Number 36417 Answers: 1 Comments: 1

calculate ∫_2 ^6 (dx/((√(x+1)) +(√(x−1))))

$${calculate}\:\int_{\mathrm{2}} ^{\mathrm{6}} \:\:\:\frac{{dx}}{\sqrt{{x}+\mathrm{1}}\:+\sqrt{{x}−\mathrm{1}}} \\ $$

Question Number 36416 Answers: 0 Comments: 0

calculate I = ∫_1 ^2 ((2x^3 +5x^2 −4x−7)/((x+2)^2 ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\frac{\mathrm{2}{x}^{\mathrm{3}} \:+\mathrm{5}{x}^{\mathrm{2}} \:−\mathrm{4}{x}−\mathrm{7}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 36415 Answers: 0 Comments: 1

calculate I = ∫_0 ^(π/2) (x^3 +x)cos^2 xdx and J = ∫_0 ^(π/2) (x^3 +x)sin^2 xdx cslculate I and J .

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({x}^{\mathrm{3}} \:+{x}\right){cos}^{\mathrm{2}} {xdx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\left({x}^{\mathrm{3}} \:+{x}\right){sin}^{\mathrm{2}} {xdx} \\ $$$${cslculate}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 36413 Answers: 0 Comments: 1

let I_n = ∫_0 ^1 x^n (√(3+x))dx 1)calculate lim_(n→+∞) I_n 2) calculate lim_(n→+∞) n I_n

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{3}+{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{I}_{{n}} \\ $$

Question Number 36412 Answers: 1 Comments: 1

calculate I_λ =∫_0 ^λ e^(−x) ln(1+e^x )dx

$${calculate}\:{I}_{\lambda} \:=\int_{\mathrm{0}} ^{\lambda} \:{e}^{−{x}} {ln}\left(\mathrm{1}+{e}^{{x}} \right){dx} \\ $$

Question Number 36411 Answers: 0 Comments: 0

calculate ∫_1 ^3 ((x−1)/(∣x^2 −2x∣ +1))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{x}−\mathrm{1}}{\mid{x}^{\mathrm{2}} −\mathrm{2}{x}\mid\:+\mathrm{1}}{dx} \\ $$

Question Number 36410 Answers: 0 Comments: 1

find the value of I_n = ∫_0 ^1 x^n (√(1−x))dx

$${find}\:{the}\:{value}\:{of}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$

Question Number 36406 Answers: 2 Comments: 1

find the values of I = ∫_0 ^π cos^4 dx and J = ∫_0 ^π sin^4 dx .

$${find}\:{the}\:{values}\:{of}\:\:{I}\:=\:\int_{\mathrm{0}} ^{\pi} {cos}^{\mathrm{4}} {dx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{4}} {dx}\:. \\ $$

Question Number 36397 Answers: 2 Comments: 0

find I = ∫_1 ^2 (dx/(x(√(x+1)) +(x+1)(√x)))

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

Question Number 36394 Answers: 0 Comments: 0

let f(x)=artanx find L(f(x)) L mean laplace trsnsform.

$${let}\:{f}\left({x}\right)={artanx}\:{find}\:\:{L}\left({f}\left({x}\right)\right) \\ $$$${L}\:{mean}\:{laplace}\:{trsnsform}. \\ $$

Question Number 36393 Answers: 0 Comments: 0

let f(x) = (2/(sinx)) ,2π periodic odd developp f at fourier serie .

$${let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{sinx}}\:\:\:,\mathrm{2}\pi\:{periodic}\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 36336 Answers: 0 Comments: 4

find f(x)= ∫_0 ^∞ arctan(xt^2 )dt with x>0

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:{arctan}\left({xt}^{\mathrm{2}} \right){dt}\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 36335 Answers: 0 Comments: 1

find f(t) = ∫_0 ^1 arctan(tx^2 )dx with t≥0 developp f at integr serie

$${find}\:\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{arctan}\left({tx}^{\mathrm{2}} \right){dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$$${developp}\:\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 36205 Answers: 0 Comments: 0

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

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

Question Number 36204 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((x^2 −1)/(x^2 +1)) ((sin(x))/x)dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} \:+\mathrm{1}}\:\frac{{sin}\left({x}\right)}{{x}}{dx}\: \\ $$

Question Number 36203 Answers: 0 Comments: 1

let f(t) = ∫_0 ^∞ ((cos(tx))/((2+x^2 )^2 ))dx 1) find a simple form of f(t) 2) calculate ∫_0 ^∞ ((cos(3x))/((2+x^2 )^2 ))dx

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({tx}\right)}{\left(\mathrm{2}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left(\mathrm{3}{x}\right)}{\left(\mathrm{2}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

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

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

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