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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→+∞)

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

Question Number 34221 Answers: 1 Comments: 1

study the convergence of ∫_0 ^1 ((√(1−x))/x) dx .

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}\:{dx}\:. \\ $$

Question Number 34220 Answers: 0 Comments: 0

calculate I = ∫_0 ^(π/4) cosx ln(tanx)dx .

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cosx}\:{ln}\left({tanx}\right){dx}\:. \\ $$

Question Number 34219 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) (dx/(cos^3 x +sin^3 x))

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dx}}{{cos}^{\mathrm{3}} {x}\:+{sin}^{\mathrm{3}} {x}} \\ $$

Question Number 34218 Answers: 0 Comments: 0

find ∫(√(tanx))dx 2) calculate ∫_0 ^(π/6) (√(tanx)) dx

$${find}\:\int\sqrt{{tanx}}{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \sqrt{{tanx}}\:{dx} \\ $$

Question Number 34217 Answers: 0 Comments: 1

calculate lim_(n→+∞) n^3 Σ_(k=1) ^n (1/(n^4 +k^2 n^2 +k^4 )) .

$${calculate}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{3}} \:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} \:+{k}^{\mathrm{2}} {n}^{\mathrm{2}} \:+{k}^{\mathrm{4}} }\:. \\ $$

Question Number 34216 Answers: 0 Comments: 0

let give I =∫_0 ^1 ((ln(x+1))/x)dx and J = ∫_0 ^1 ((ln(1−x))/x)dx 1) prove the existence of I and J 2) calculate I +J and 2I +J 3) find I and J .

$${let}\:{give}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{I}\:{and}\:{J} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:+{J}\:{and}\:\mathrm{2}{I}\:+{J} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 34215 Answers: 0 Comments: 0

find the polynome p_n wich verify p_n (0)=0 and ∀ x ∈ R p_n (x)−p_n (x−1) =x^n

$${find}\:{the}\:{polynome}\:{p}_{{n}} \:{wich}\:{verify}\:{p}_{{n}} \left(\mathrm{0}\right)=\mathrm{0}\:{and} \\ $$$$\forall\:{x}\:\in\:{R}\:\:{p}_{{n}} \left({x}\right)−{p}_{{n}} \left({x}−\mathrm{1}\right)\:={x}^{{n}} \\ $$

Question Number 34211 Answers: 1 Comments: 0

let x and y such that 2x^2 +4x−2y=0 y^2 −(x+6)^2 =0 find the possibles value of x+y

$${let}\:{x}\:{and}\:{y}\:{such}\:{that} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{2}{y}=\mathrm{0} \\ $$$${y}^{\mathrm{2}} −\left({x}+\mathrm{6}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:{the}\:{possibles}\:{value}\:{of}\:{x}+{y} \\ $$

Question Number 34208 Answers: 0 Comments: 0

the 15^(th) term of an AP is 34 the sum of the 94^(th) to 100^(th) term is 2000 find the 2^(nd) term and the mean from the sixth term to the 10^(th) term if the 100^(th) term is 360.

$$\:{the}\:\mathrm{15}^{{th}} \:{term}\:{of}\:{an}\:{AP}\:{is}\:\mathrm{34}\:{the}\: \\ $$$${sum}\:{of}\:{the}\:\mathrm{94}^{{th}} \:{to}\:\mathrm{100}^{{th}} \:{term}\:{is} \\ $$$$\mathrm{2000}\:{find}\:{the}\:\mathrm{2}^{{nd}} \:{term}\:{and}\:{the} \\ $$$${mean}\:{from}\:{the}\:{sixth}\:{term}\:{to}\:{the} \\ $$$$\mathrm{10}^{{th}} \:{term}\:{if}\:{the}\:\mathrm{100}^{{th}} \:{term}\:{is} \\ $$$$\:\mathrm{360}. \\ $$

Question Number 34205 Answers: 1 Comments: 1

Question Number 34206 Answers: 0 Comments: 1

Question Number 34202 Answers: 1 Comments: 0

find the gradient of the curve y=(1/x).

$$\:\:{find}\:{the}\:{gradient}\:{of}\:{the}\:{curve} \\ $$$$\:\:\:\:{y}=\frac{\mathrm{1}}{{x}}. \\ $$

Question Number 34196 Answers: 1 Comments: 2

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