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IntegrationQuestion and Answers: Page 243

Question Number 60621 Answers: 0 Comments: 5

if π is rational then there exists a I_n =(v^(2n) /(n!))∫_0 ^π x^n (x−π)^n sin(x)dx can someone give a easier way to expaned this

$${if}\:\pi\:{is}\:{rational}\:{then}\:{there} \\ $$$${exists}\:{a}\:{I}_{{n}} =\frac{{v}^{\mathrm{2}{n}} }{{n}!}\underset{\mathrm{0}} {\overset{\pi} {\int}}{x}^{{n}} \left({x}−\pi\right)^{{n}} {sin}\left({x}\right){dx} \\ $$$${can}\:{someone}\:{give}\:{a}\:{easier}\:{way}\:{to}\:{expaned}\:{this} \\ $$

Question Number 60631 Answers: 0 Comments: 0

prove that∫_(−∞) ^∞ x^5 e^(−x^2 ) sin(x^3 ) dx=0.25474

$$\mathrm{prove}\:\mathrm{that}\underset{−\infty} {\overset{\infty} {\int}}\mathrm{x}^{\mathrm{5}} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{sin}\left(\mathrm{x}^{\mathrm{3}} \right)\:\mathrm{dx}=\mathrm{0}.\mathrm{25474} \\ $$

Question Number 60586 Answers: 0 Comments: 1

find ∫_0 ^1 ((ln^2 (x))/((1−x^2 )^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 60534 Answers: 0 Comments: 1

Question Number 60506 Answers: 0 Comments: 1

calculate ∫∫_W ((√(2x^2 +3y^2 ))/(x+y)) dxdy with W ={(x,y)∈R^2 / 0<x<1 and 0<y<1.

$${calculate}\:\int\int_{{W}} \:\:\:\:\:\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }}{{x}+{y}}\:{dxdy} \\ $$$${with}\:{W}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}<{x}<\mathrm{1}\:{and}\:\mathrm{0}<{y}<\mathrm{1}.\right. \\ $$

Question Number 60498 Answers: 0 Comments: 4

let f(t) =∫_0 ^3 (√(t +x +x^2 ))dx with t ≥(1/4) 1) find a explicit form of f(t) 2) find also g(t) = ∫_0 ^3 (dx/(√(t+x +x^2 ))) 3) calculate ∫_0 ^3 (√(1+x+x^2 ))dx , ∫_0 ^3 (√(2 +x+x^2 ))dx ∫_0 ^3 (dx/(√(2+x +x^2 ))) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{{t}\:+{x}\:+{x}^{\mathrm{2}} }{dx}\:\:{with}\:{t}\:\geqslant\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \:\:\:\frac{{dx}}{\sqrt{{t}+{x}\:+{x}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{3}} \:\sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }{dx}\:,\:\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{2}\:+{x}+{x}^{\mathrm{2}} }{dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{3}} \:\:\:\frac{{dx}}{\sqrt{\mathrm{2}+{x}\:+{x}^{\mathrm{2}} }}\:\:. \\ $$

Question Number 60595 Answers: 0 Comments: 2

let f(a) =∫_0 ^1 ((ln^2 (x))/((1−ax)^2 )) dx with ∣a∣<1 1) find a explicit form of f(a) 2) determine A(θ) =∫_0 ^1 ((ln^2 (x))/((1−(cosθ)x)^2 ))dx with 0<θ<(π/2)

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−{ax}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−\left({cos}\theta\right){x}\right)^{\mathrm{2}} }{dx}\:\:{with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 60496 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((lnx)/((1−x)^2 ))dx

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

Question Number 60495 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(x))/((1−x^2 )^2 ))dx

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

Question Number 60494 Answers: 1 Comments: 1

find ∫ (√((√(2+x^2 ))−x))dx

$${find}\:\int\:\sqrt{\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }−{x}}{dx} \\ $$

Question Number 60481 Answers: 0 Comments: 0

Question Number 60424 Answers: 0 Comments: 2

let z ∈C and ∣z∣<1 find f(x)=∫_0 ^1 ln(1+zx)dx.

$${let}\:{z}\:\in{C}\:{and}\:\:\mid{z}\mid<\mathrm{1}\:\:{find} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{zx}\right){dx}. \\ $$

Question Number 60413 Answers: 1 Comments: 4

Question Number 60384 Answers: 0 Comments: 0

∫e^(coth^(−1) (x)) dx

$$\int{e}^{{coth}^{−\mathrm{1}} \left({x}\right)} \:{dx}\: \\ $$

Question Number 60376 Answers: 1 Comments: 2

Question Number 60335 Answers: 0 Comments: 0

find I_n = ∫ x^n arctan(x)dx with n integr natural.

$${find}\:{I}_{{n}} =\:\int\:\:{x}^{{n}} \:{arctan}\left({x}\right){dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$

Question Number 60346 Answers: 0 Comments: 1

∫xsec^3 xdx please help

$$\int{x}\mathrm{sec}\:^{\mathrm{3}} {xdx} \\ $$$${please}\:{help} \\ $$

Question Number 60320 Answers: 0 Comments: 1

Question Number 60319 Answers: 0 Comments: 0

Question Number 60318 Answers: 0 Comments: 2

Question Number 60311 Answers: 1 Comments: 2

∫(dx/(√(sec h^2 (x)+1))) dx

$$\int\frac{{dx}}{\sqrt{{sec}\:{h}^{\mathrm{2}} \left({x}\right)+\mathrm{1}}}\:{dx} \\ $$

Question Number 60264 Answers: 0 Comments: 0

let f(t) =∫_0 ^∞ (e^(−3 [x^2 ]) /(x^2 +t^2 ))dx with t>0 1. determine a explicit form of f(t) 2. find also g(t) =∫_0 ^∞ (e^(−3[x^2 ]) /((x^2 +t^2 )^2 ))dx 3. find the values of integrals ∫_0 ^∞ (e^(−3[x^2 ]) /(x^2 +3))dx and ∫_0 ^∞ (e^(−3[x^2 ]) /((x^2 +4)^2 )) dx .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−\mathrm{3}\:\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dx}\:\:{with}\:{t}>\mathrm{0} \\ $$$$\mathrm{1}.\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\mathrm{2}.\:{find}\:{also}\:{g}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\mathrm{3}.\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 60263 Answers: 0 Comments: 1

let U_n =∫_0 ^∞ (e^(−n[x^2 ]) /(x^2 +3)) dx 1) calculate U_n interms of n 2) find lim_(n→+∞) n U_n 3)determine nature of the serie Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{n}\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} +\mathrm{3}}\:{dx}\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right){determine}\:{nature}\:{of}\:{the}\:{serie}\:\:\Sigma\:{U}_{{n}} \\ $$

Question Number 60234 Answers: 1 Comments: 0

Question Number 60170 Answers: 0 Comments: 0

∫xsec^3 xdx

$$\int{x}\mathrm{sec}\:^{\mathrm{3}} {xdx} \\ $$

Question Number 60050 Answers: 1 Comments: 1

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

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

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