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Question Number 33352 Answers: 0 Comments: 1

let give S(x)=Σ_(n≥0) (((−1)^n )/(√(x+n))) ,x>0 1)study the contnuity ,derivsbility,limits at 0^+ and +∞ 2) we give ∫_0 ^∞ e^(−t^2 ) dt =((√π)/2) .prove that ∀ x>0 S(x)=(1/(√π)) ∫_0 ^∞ (e^(−tx) /((√t)(1+e^(−t) )))dt .

$${let}\:{give}\:{S}\left({x}\right)=\sum_{{n}\geqslant\mathrm{0}} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\sqrt{{x}+{n}}}\:,{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{contnuity}\:,{derivsbility},{limits} \\ $$$${at}\:\mathrm{0}^{+} \:{and}\:+\infty \\ $$$$\left.\mathrm{2}\right)\:{we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:.{prove}\:{that} \\ $$$$\forall\:{x}>\mathrm{0}\:\:{S}\left({x}\right)=\frac{\mathrm{1}}{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}} }{\sqrt{{t}}\left(\mathrm{1}+{e}^{−{t}} \right)}{dt}\:. \\ $$

Question Number 33351 Answers: 0 Comments: 1

prove that ∫_0 ^1 (((−lnx)^p )/(1+x^2 )) =p! Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^(p+1) )) p integr.

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(−{lnx}\right)^{{p}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:={p}!\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{{p}+\mathrm{1}} } \\ $$$${p}\:{integr}. \\ $$

Question Number 33350 Answers: 0 Comments: 1

prove that ∫_0 ^∞ ((cos(αx))/(chx))dx= 2 Σ_(n=0) ^∞ (−1)^n ((2n+1)/((2n+1)^2 +α^2 )) .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\alpha{x}\right)}{{chx}}{dx}= \\ $$$$\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\frac{\mathrm{2}{n}+\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} \:+\alpha^{\mathrm{2}} }\:. \\ $$

Question Number 33349 Answers: 0 Comments: 1

prove that ∫_0 ^∞ x(x−ln(e^x −1))dx=Σ_(n=1) ^∞ (1/n^3 )

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{x}\left({x}−{ln}\left({e}^{{x}} −\mathrm{1}\right)\right){dx}=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} } \\ $$

Question Number 33346 Answers: 0 Comments: 0

let a and b from R /a<b f [a,b]→C continje prove that ∀n ∈N ∫_a ^b (Π_(k=0) ^(n−1) (x+k))f(x)dx=0 ⇒ f=0

$${let}\:{a}\:{and}\:{b}\:{from}\:{R}\:/{a}<{b}\:\:{f}\:\:\left[{a},{b}\right]\rightarrow{C}\:{continje} \\ $$$${prove}\:{that}\:\:\forall{n}\:\in{N}\:\:\int_{{a}} ^{{b}} \left(\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({x}+{k}\right)\right){f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\Rightarrow\:{f}=\mathrm{0} \\ $$

Question Number 33345 Answers: 0 Comments: 0

for x∈]0,+∞[ let ψ(x) = ((Γ^′ (x))/(Γ(x))) 1)prove that ψ(x) =−(1/x) −γ +x Σ_(n=1) ^∞ (1/(n(x+n))) 2)ptove that γ =−Γ^′ (1) 3) prove that ∫_0 ^∞ e^(−x) ln(x)dx =−γ .

$$\left.{for}\:{x}\in\right]\mathrm{0},+\infty\left[\:{let}\:\psi\left({x}\right)\:=\:\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)}\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\psi\left({x}\right)\:=−\frac{\mathrm{1}}{{x}}\:−\gamma\:+{x}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}\left({x}+{n}\right)} \\ $$$$\left.\mathrm{2}\right){ptove}\:{that}\:\gamma\:=−\Gamma^{'} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:. \\ $$

Question Number 33344 Answers: 0 Comments: 1

prove that ∀ α ∈]0,+∞[ lim_(n→∞) ∫_0 ^n (1−(x/n))^n x^(α−1) dx =Γ(α) .

$$\left.{prove}\:{that}\:\:\forall\:\alpha\:\in\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \:{x}^{\alpha−\mathrm{1}} {dx}\:=\Gamma\left(\alpha\right)\:. \\ $$

Question Number 33343 Answers: 0 Comments: 2

prove that ∀ α ∈]1,+∞[ lim_(n→∞) ∫_0 ^n (1+(x/n))^n e^(−αx) dx = (1/(α−1)) .

$$\left.{prove}\:{that}\:\forall\:\alpha\:\in\right]\mathrm{1},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} \:{e}^{−\alpha{x}} {dx}\:=\:\frac{\mathrm{1}}{\alpha−\mathrm{1}}\:. \\ $$

Question Number 33342 Answers: 0 Comments: 4

let I_n = ∫_0 ^1 (1/x)(1−(1−(x/n))^n )dx and J_n = ∫_1 ^n (1/x)(1−(x/n))^n dx ,n integr not 0 1) prove that lim_ I_n =∫_0 ^1 ((1−e^(−x) )/x)dx lim J_n = ∫_0 ^1 (e^(−(1/x)) /x) dx (n→∞) 2) prove that ∀n∈ N^★ I_n −J_n = Σ_(k=1) ^n (1/k) −ln(n) 3) prove that ∫_0 ^1 ((1−e^(−x) −e^(−(1/x)) )/x) =γ

$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \right){dx}\:{and} \\ $$$${J}_{{n}} \:=\:\int_{\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx}\:\:,{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{lim}_{} \:{I}_{{n}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−{e}^{−{x}} }{{x}}{dx} \\ $$$${lim}\:{J}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−\frac{\mathrm{1}}{{x}}} }{{x}}\:{dx}\:\left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\forall{n}\in\:{N}^{\bigstar} \:\:\:{I}_{{n}} \:−{J}_{{n}} \:=\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\mathrm{1}−{e}^{−{x}} \:−{e}^{−\frac{\mathrm{1}}{{x}}} }{{x}}\:=\gamma \\ $$

Question Number 33341 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )(a^2 +x^2 ))) 2) find the value of A(θ) = ∫_0 ^∞ (dx/((1+x^2 )( x^2 +1 −sin^2 θ))) 0<θ<(π/2) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of} \\ $$$${A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:{x}^{\mathrm{2}} \:+\mathrm{1}\:−{sin}^{\mathrm{2}} \theta\right)} \\ $$$$\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:. \\ $$

Question Number 33340 Answers: 0 Comments: 1

find a equivalent of A_n = ∫_0 ^n (√(1 +(1−(x/n))^n )) dx (n→∞)

$${find}\:{a}\:{equivalent}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:\sqrt{\mathrm{1}\:+\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} }\:\:{dx}\:\left({n}\rightarrow\infty\right) \\ $$

Question Number 33339 Answers: 0 Comments: 1

find lim_(n→∞) ∫_0 ^∞ e^(−x^2 ) sin^n xdx

$${find}\:\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {xdx}\: \\ $$

Question Number 33338 Answers: 0 Comments: 1

find lim_(n→∞) ∫_0 ^(+∞) tan^n xdx .