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let give a>0 1) find the value of F(a) = ∫_0 ^∞ ((lnt)/(t^2 +a^2 ))dt 2) find the value of G(a)=∫_0 ^∞ ((aln(t))/((t^2 +a^2 )^2 ))dt 3) find the value of ∫_0 ^∞ ((ln(t))/((t^2 +3)^2 ))dt

$${let}\:{give}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{F}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnt}}{{t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{G}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{aln}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 34258 Answers: 0 Comments: 1

1) prove that ∀ x∈]0,1[ 1−(1/x)≤lnx≤ x−1 2) find 2 sequences u_n and v_n / u_n ≤Π_(k=1) ^(n−1) ln((k/n))≤v_n ∀n≥2

$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:\forall\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:\mathrm{1}−\frac{\mathrm{1}}{{x}}\leqslant{lnx}\leqslant\:{x}−\mathrm{1}\right. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\mathrm{2}\:{sequences}\:{u}_{{n}} \:{and}\:{v}_{{n}} \:\:\:/ \\ $$$${u}_{{n}} \leqslant\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{ln}\left(\frac{{k}}{{n}}\right)\leqslant{v}_{{n}} \:\:\:\:\forall{n}\geqslant\mathrm{2} \\ $$

Question Number 34257 Answers: 0 Comments: 0

find f(x)= ∫_1 ^x (dt/(t(√(1+t^2 )))) 2) calculate I =∫_1 ^(+∞) (dt/(t(√(1+t^2 ))))

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{1}} ^{{x}} \:\:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:=\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$

Question Number 34255 Answers: 0 Comments: 0

find g(x)= ∫_0 ^∞ ((ln(1+xt^2 ))/t^2 ) dt 2) calculate ∫_0 ^∞ ((ln(1+3t^2 ))/t^2 )dt .

$${find}\:\:{g}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+\mathrm{3}{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:. \\ $$

Question Number 34254 Answers: 0 Comments: 0

find I(x)= ∫_0 ^1 ((ln(1+xt^2 ))/t^2 )dt .

$${find}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:. \\ $$

Question Number 34253 Answers: 0 Comments: 0

let F(x)= ∫_0 ^x ((ln(1+t^2 ))/t^2 )dt 1) calculate F(x) 2) find the value of ∫_0 ^∞ ((ln(1+t^2 ))/t^2 )dt

$$\:{let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 34326 Answers: 1 Comments: 0

find the equation of the 2D curve such that the lines (x/t) + (y/((a−t) )) = 1 are always tangent to the curve. given ′a′ is a positive real constant and ′t′ is a parameter. ( 0 < t < a )

$${find}\:{the}\:{equation}\:{of}\:{the}\:\mathrm{2}{D} \\ $$$${curve}\:{such}\:{that}\:{the}\:{lines} \\ $$$$\:\frac{{x}}{{t}}\:+\:\frac{{y}}{\left({a}−{t}\right)\:}\:=\:\mathrm{1} \\ $$$$\:{are}\:{always}\:{tangent}\:{to} \\ $$$${the}\:{curve}. \\ $$$${given}\:'{a}'\:\:{is}\:{a}\:{positive}\:{real} \\ $$$${constant}\:{and}\:'{t}'\:{is}\:{a} \\ $$$${parameter}.\:\left(\:\mathrm{0}\:<\:{t}\:<\:{a}\:\right) \\ $$

Question Number 34237 Answers: 0 Comments: 4

find ∫ (dx/(x^2 −a)) with a ∈ C .

$${find}\:\:\int\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−{a}}\:\:{with}\:{a}\:\in\:{C}\:. \\ $$

Question Number 34231 Answers: 1 Comments: 0

Question Number 34230 Answers: 1 Comments: 0

Question Number 34229 Answers: 2 Comments: 3

calculate ∫_(−∞) ^∞ ((cos(tx))/(1+x^4 )) dx with t≥0 2) calculate ∫_0 ^∞ (dx/(1+x^4 )) .

$${calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{{cos}\left({tx}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} }\:. \\ $$

Question Number 34228 Answers: 0 Comments: 1

find the value of ∫_0 ^1 (x^2 /(1+x^4 ))dx

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

Question Number 34227 Answers: 1 Comments: 2

calculate ∫_0 ^1 arctan(x^2 )dx

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

Question Number 34226 Answers: 0 Comments: 1

let u_n = (n+1)^((n+1)/n) −n^(n/(n−1)) find lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\left({n}+\mathrm{1}\right)^{\frac{{n}+\mathrm{1}}{{n}}} \:\:−{n}^{\frac{{n}}{{n}−\mathrm{1}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$

Question Number 34225 Answers: 1 Comments: 0

find ∫ (dx/(1+x^2 +x^4 ))

$${find}\:\int\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} } \\ $$

Question Number 34224 Answers: 0 Comments: 0

study and give the graph for f(x) =(((x−1)^2 )/x) e^(−(1/(x^2 ))) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:\:{for} \\ $$$${f}\left({x}\right)\:=\frac{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }{{x}}\:{e}^{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} \:\:}} . \\ $$

Question Number 34223 Answers: 0 Comments: 0

find ∫ (dx/(x^(2n) −1)) with n integr natural and n≥1 .

$${find}\:\int\:\:\:\frac{{dx}}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 34222 Answers: 0 Comments: 4

let give the sequence of integrals J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx 1) prove that J_n =(n−1)J_(n−2) ∀n≥2 2) calculate J_(2p) and J_(2p+1) by using factoriels. 3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) . 4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p))) 5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)

$${let}\:{give}\:{the}\:{sequence}\:{of}\:{integrals} \\ $$$${J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{{n}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} {dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{J}_{{n}} =\left({n}−\mathrm{1}\right){J}_{{n}−\mathrm{2}} \:\:\:\forall{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{J}_{\mathrm{2}{p}} \:{and}\:{J}_{\mathrm{2}{p}+\mathrm{1}} \:{by}\:{using}\:{factoriels}. \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\forall{n}\geqslant\mathrm{1}\:\:\:{J}_{{n}} ^{\mathrm{2}} \:\:\leqslant{J}_{{n}−\mathrm{1}} \:.\:{J}_{{n}+\mathrm{1}} . \\ $$$$\left.\mathrm{4}\right){prove}\:{that}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}+\mathrm{1}}}\:\leqslant{J}_{\mathrm{0}} \:\leqslant\:\frac{\mathrm{2}^{\mathrm{2}{p}} \:\left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}}} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{a}\:{equivalent}\:{of}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\:\left({p}\rightarrow+\infty\right) \\ $$

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