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IntegrationQuestion and Answers: Page 235

Question Number 62812 Answers: 0 Comments: 1

let U_n =∫_0 ^(+∞) ((arctan(nt))/(1+n^2 t^2 ))dt with n natural≥1 1) calculate U_n 2) calculate lim_(n→+∞) n^2 U_n 3) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left({nt}\right)}{\mathrm{1}+{n}^{\mathrm{2}} {t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:{n}\:{natural}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 62811 Answers: 0 Comments: 2

1) find ∫ ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx 2)calculate ∫_5 ^(+∞) ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx

$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$

Question Number 62808 Answers: 0 Comments: 0

f(t) =∫_0 ^(+∞) (e^(−xt) /((x+t)^2 ))dx with t≥0 1) study the set of definition for f(t) 2)study the continuity of f 3)study the derivability of f 4) developp f at integr serie

$${f}\left({t}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{xt}} }{\left({x}+{t}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{set}\:{of}\:{definition}\:{for}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{derivability}\:{of}\:{f} \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 62806 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((x+1)/((x^4 +x^2 +1)^3 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 62805 Answers: 0 Comments: 1

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

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

Question Number 62790 Answers: 1 Comments: 1

Question Number 62781 Answers: 0 Comments: 0

∫ sec(2x) e^(2x) dx

$$\int\:\mathrm{sec}\left(\mathrm{2x}\right)\:\mathrm{e}^{\mathrm{2x}} \:\:\mathrm{dx} \\ $$

Question Number 62732 Answers: 1 Comments: 0

calculate ∫_0 ^(2π) ((cos(2x))/(2cosx −sin(x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}{cosx}\:−{sin}\left({x}\right)}{dx}\: \\ $$

Question Number 62731 Answers: 0 Comments: 1

find ∫ (√((x−1)/(x^2 +3)))dx

$${find}\:\int\:\sqrt{\frac{{x}−\mathrm{1}}{{x}^{\mathrm{2}} \:+\mathrm{3}}}{dx}\: \\ $$

Question Number 62653 Answers: 1 Comments: 4

∫x(arctan(x))^2 dx ∫((x e^(arctan(x)) )/((1+x^2 )^(3/2) )) dx ∫((arcsin(x))/(√(1+x))) dx

$$\int\mathrm{x}\left(\mathrm{arctan}\left(\mathrm{x}\right)\right)^{\mathrm{2}} \:\mathrm{dx} \\ $$$$ \\ $$$$\int\frac{\mathrm{x}\:\mathrm{e}^{\mathrm{arctan}\left(\mathrm{x}\right)} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\mathrm{dx} \\ $$$$ \\ $$$$\int\frac{\mathrm{arcsin}\left(\mathrm{x}\right)}{\sqrt{\mathrm{1}+\mathrm{x}}}\:\mathrm{dx} \\ $$

Question Number 62648 Answers: 0 Comments: 1

Question Number 62613 Answers: 3 Comments: 2

Question Number 62596 Answers: 1 Comments: 0

∫sin^(100) (x) cos^(100) (x) dx

$$\int\mathrm{sin}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{cos}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 62455 Answers: 0 Comments: 3

Question Number 62453 Answers: 0 Comments: 3

∫ (x/(e^x − 1))dx, for x > 0

$$\int\:\frac{\mathrm{x}}{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 62437 Answers: 0 Comments: 1

let f(x) =∫_0 ^1 ((arctan(1+xt))/(t^2 +1))dt determine a explicit form for f(x) 2)calculate ∫_0 ^1 ((arctan(1+2t))/(1+t^2 ))dt

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+{xt}\right)}{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt} \\ $$$${determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{1}+\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$ \\ $$

Question Number 62425 Answers: 0 Comments: 0

let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) with x>1 1) calculate lim_(x→1^+ ) ξ(x) and lim_(x→+∞) ξ(x) 2) prove that ξ(x) =1+2^(−x) +o(2^(−x) ) (x→+∞) 3) prove that ξ is decreasing and convexe fucntion on]1,+∞[

$${let}\:\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{with}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \:\:\xi\left({x}\right)\:\:{and}\:{lim}_{{x}\rightarrow+\infty} \:\:\:\xi\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\xi\left({x}\right)\:=\mathrm{1}+\mathrm{2}^{−{x}} \:+{o}\left(\mathrm{2}^{−{x}} \right)\:\:\:\left({x}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\xi\:{is}\:{decreasing}\:{and}\:{convexe}\:{fucntion}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$

Question Number 62420 Answers: 0 Comments: 0

let u_n (x)=(1/n^x ) −∫_n ^(n+1) (dt/t^x ) with x∈[1,2] 1)prove that 0≤ u_n (x)≤(1/n^x )−(1/((n+1)^x )) (n>0) 2)prove that Σ u_n (x)converges let γ =Σ_(n=1) ^∞ u_n (1) 3)find Σ_(n=1) ^∞ u_n (x) interms of ξ(x)and 1−x 4) prove that the converg.of Σu_n (x)is uniform prove that for x∈V(1) ξ(x) =(1/(x−1)) +γ +o(1) 5) find the value of Σ_(n=1) ^∞ (((−1)^(n−1) )/n)ln(n)

$${let}\:{u}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{{n}^{{x}} }\:−\int_{{n}} ^{{n}+\mathrm{1}} \frac{{dt}}{{t}^{{x}} }\:\:{with}\:{x}\in\left[\mathrm{1},\mathrm{2}\right] \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\mathrm{0}\leqslant\:{u}_{{n}} \left({x}\right)\leqslant\frac{\mathrm{1}}{{n}^{{x}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{{x}} }\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\:{u}_{{n}} \left({x}\right){converges} \\ $$$${let}\:\gamma\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{u}_{{n}} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right){find}\:\sum_{{n}=\mathrm{1}} ^{\infty} {u}_{{n}} \left({x}\right)\:{interms}\:{of}\:\xi\left({x}\right){and} \\ $$$$\mathrm{1}−{x} \\ $$$$\left.\mathrm{4}\right)\:{prove}\:{that}\:{the}\:{converg}.{of}\:\Sigma{u}_{{n}} \left({x}\right){is} \\ $$$${uniform} \\ $$$${prove}\:{that}\:{for}\:{x}\in{V}\left(\mathrm{1}\right) \\ $$$$\xi\left({x}\right)\:=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\gamma\:+{o}\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{5}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}{ln}\left({n}\right) \\ $$

Question Number 62419 Answers: 0 Comments: 1

calculate ∫_0 ^1 (2x^2 −1)(√(x^2 −2x+5))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$

Question Number 62418 Answers: 0 Comments: 0

calculate ∫_0 ^1 Γ(t).Γ(1−t)dt

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\Gamma\left({t}\right).\Gamma\left(\mathrm{1}−{t}\right){dt}\: \\ $$

Question Number 62417 Answers: 0 Comments: 0

prove that Γ(x).Γ(1−x) =(π/(sin(πx))) with 0<x<1

$${prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\:\:\:\:{with}\:\mathrm{0}<{x}<\mathrm{1} \\ $$

Question Number 62416 Answers: 0 Comments: 1

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>1 calculate Γ^((n)) (x) for all integr n.