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IntegrationQuestion and Answers: Page 313

Question Number 30773 Answers: 0 Comments: 1

let a>0 calculate ∫_0 ^∞ (dx/((x^2 +a^2 )^2 )) 2) calculate ∫_(−∞) ^(+∞) (x^2 /((x^2 +a^2 )^2 ))dx.

$${let}\:{a}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30769 Answers: 0 Comments: 1

find the value of I= ∫_0 ^1 (dx/((x+1)^2 (√(x^2 +2x +2)))) .

$${find}\:{the}\:{value}\:{of}\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{2}}}\:. \\ $$

Question Number 30767 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ((1−(√x))/(1−^n (√x)))dx.

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−^{{n}} \sqrt{{x}}}{dx}. \\ $$

Question Number 30766 Answers: 0 Comments: 0

find ∫_0 ^1 (x/(√(x^4 +x^2 +1)))dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{x}}{\sqrt{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}}{dx} \\ $$

Question Number 30765 Answers: 0 Comments: 1

find ∫_0 ^1 (dx/(√(x^2 +x+1))) .

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}}\:. \\ $$

Question Number 30764 Answers: 0 Comments: 1

let I_n = ∫_0 ^(1/2) (1−2t)^n e^(−t) dt with n integr not 0 1) prove that ∀t∈[0,(1/2)] (1/(√e))(1−2t)^n ≤ (1−2t)^n e^(−t) ≤(1−2t)^n then find lim_(n→∞ ) I_n 2) prove that I_(n+1 ) =1−2(n+1)I_n 3) calculate I_1 ,I_2 , and I_3 .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{I}_{{n}+\mathrm{1}\:} =\mathrm{1}−\mathrm{2}\left({n}+\mathrm{1}\right){I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,\:{and}\:{I}_{\mathrm{3}} . \\ $$

Question Number 30761 Answers: 0 Comments: 1

find ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx with n from N.

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:\:\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 30760 Answers: 0 Comments: 1

find I_n = ∫_0 ^1 (lnx)^n dx with n fromN

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({lnx}\right)^{{n}} \:{dx}\:\:{with}\:{n}\:{fromN} \\ $$

Question Number 30741 Answers: 0 Comments: 0

let give D= R_+ ^2 −{(0,0)} and α from R let C_1 ={(x,y)∈ D/0<x^2 +y^2 ≤1 } C_2 ={(x,y) ∈D / x^2 +y^2 ≥1} study the convergence of I= ∫∫_C_1 ((dxdy)/(((√(x^2 +y^2 )) )^α )) and J=∫∫_C_2 ((dxdy)/(((√(x^2 +y^2 )) )^α )) .

$${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}} } \:\:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\right)^{\alpha} }\:\:{and}\:{J}=\int\int_{{C}_{\mathrm{2}} } \:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:}\:\right)^{\alpha} }\:. \\ $$

Question Number 30737 Answers: 0 Comments: 1

∫(1/(x^2 +ln x))dx

$$\int\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{ln}\:{x}}{dx} \\ $$

Question Number 30665 Answers: 0 Comments: 0

find ∫_0 ^π (dx/(1+cos(2x) +sin(2x))) .

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)\:+{sin}\left(\mathrm{2}{x}\right)}\:. \\ $$

Question Number 30585 Answers: 0 Comments: 0

find F_n (x)= ∫_0 ^∞ (x^n /(e^(x+n) +1))dx .

$${find}\:\:{F}_{{n}} \left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{{n}} }{{e}^{{x}+{n}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 30584 Answers: 0 Comments: 0

find I= ∫_(−∞) ^(+∞) (e^(−x^2 ) /(a^2 +(v−x)^2 ))dx.

$${find}\:\:{I}=\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{e}^{−{x}^{\mathrm{2}} } }{{a}^{\mathrm{2}} \:+\left({v}−{x}\right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30580 Answers: 0 Comments: 1

decompose inside C[x] F= (x^n /(x^m +1)) with m≥n+2 then find ∫_0 ^∞ (x^n /(x^m +1))dx.

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}\:{with}\:{m}\geqslant{n}+\mathrm{2} \\ $$$${then}\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}{dx}. \\ $$

Question Number 30575 Answers: 0 Comments: 0

find ∫∫_D (x^2 +y^2 )dxdy with D={(x,y)/ x≤1 and x^2 ≤y≤2 }.

$${find}\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}=\left\{\left({x},{y}\right)/\:{x}\leqslant\mathrm{1}\:{and}\:{x}^{\mathrm{2}} \leqslant{y}\leqslant\mathrm{2}\:\right\}. \\ $$

Question Number 30574 Answers: 0 Comments: 0

find ∫∫_([1,e]^2 ) ln(xy)dxdy.

$${find}\:\int\int_{\left[\mathrm{1},{e}\right]^{\mathrm{2}} } \:\:\:{ln}\left({xy}\right){dxdy}. \\ $$

Question Number 30573 Answers: 0 Comments: 0

find ∫∫_([0,1]×[0,1]) (x^2 /(1+y^2 ))dxdy.

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right]} \:\:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{y}^{\mathrm{2}} }{dxdy}. \\ $$

Question Number 30572 Answers: 0 Comments: 1

find I=∫∫_([3,4]×[1,2]) ((dxdy)/((x+y)^2 )) .

$${find}\:\:{I}=\int\int_{\left[\mathrm{3},\mathrm{4}\right]×\left[\mathrm{1},\mathrm{2}\right]} \:\:\frac{{dxdy}}{\left({x}+{y}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 30570 Answers: 0 Comments: 0

find ∫∫_U ((dxdy)/(x^2 +y^2 )) with U= {(x,y)∈R^2 /1≤x^2 +2y^2 ≤4}

$${find}\:\int\int_{{U}} \:\frac{{dxdy}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:{with}\:{U}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{1}\leqslant{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \leqslant\mathrm{4}\right\} \\ $$

Question Number 30569 Answers: 0 Comments: 0

find I= ∫∫_D (√(1−(x^2 /a^2 )−(y^2 /b^2 ))) dxdy with D is the interior of ellipce (x^2 /a^2 ) +(y^2 /b^2 ) =1.

$${find}\:{I}=\:\int\int_{{D}} \:\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }}\:\:{dxdy}\:\:{with}\:{D}\:{is}\:{the}\:{interior} \\ $$$${of}\:{ellipce}\:\:\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\mathrm{1}. \\ $$

Question Number 30568 Answers: 0 Comments: 0

find ∫_(−1) ^1 (dx/((√(1+x^2 )) +(√(1−x^2 )))) .

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

Question Number 30566 Answers: 0 Comments: 0

study the convergence of A(α) =∫_0 ^∞ (t^(α−1) /(1+t^2 ))dt and find its value.

$${study}\:{the}\:{convergence}\:{of}\:{A}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{\alpha−\mathrm{1}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and} \\ $$$${find}\:{its}\:{value}. \\ $$

Question Number 30564 Answers: 0 Comments: 0

f and g are 2 function C^n on [a,b] prove that ∫_a ^b f^((n)) (x)g(x)dx=[Σ_(k=0) ^(n−1) (−1)^k f^((k)) g^((n−k)) ]_a ^b +(−1)^n ∫_a ^b f(x)g^((n)) (x)dx

$${f}\:{and}\:{g}\:{are}\:\mathrm{2}\:{function}\:\:{C}^{{n}} \:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\int_{{a}} ^{{b}} \:{f}^{\left({n}\right)} \left({x}\right){g}\left({x}\right){dx}=\left[\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(−\mathrm{1}\right)^{{k}} \:{f}^{\left({k}\right)} {g}^{\left({n}−{k}\right)} \right]_{{a}} ^{{b}} \:+\left(−\mathrm{1}\right)^{{n}} \int_{{a}} ^{{b}} {f}\left({x}\right){g}^{\left({n}\right)} \left({x}\right){dx} \\ $$

Question Number 30559 Answers: 0 Comments: 0

find I_n = ∫_0 ^1 (1−t^2 )^n dt .

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:. \\ $$

Question Number 30557 Answers: 0 Comments: 0

if ϕ convexe and f continue on [a,b] prove that ϕ( (1/(b−a)) ∫_a ^b f(t)dt)≤ (1/(b−a)) ∫_a ^b ϕof(t)dt.

$${if}\:\varphi\:{convexe}\:{and}\:{f}\:{continue}\:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\varphi\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}\left({t}\right){dt}\right)\leqslant\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:\varphi{of}\left({t}\right){dt}. \\ $$

Question Number 30555 Answers: 0 Comments: 0

find ∫_0 ^1 (dt/(√(1−t^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{t}^{\mathrm{4}} }}\:. \\ $$

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