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Question Number 33941 Answers: 0 Comments: 0

Question Number 33940 Answers: 3 Comments: 0

Let a,b are positive real numbers such that a−b=10 , then the smallest value of the constant k for which (√((x^2 +ax))) − (√((x^2 +bx))) < k for all x>0 is ?

$$\boldsymbol{{L}}{et}\:{a},{b}\:{are}\:{positive}\:{real}\:{numbers}\:{such} \\ $$$${that}\:{a}−{b}=\mathrm{10}\:,\:{then}\:{the}\:{smallest}\:{value} \\ $$$${of}\:{the}\:{constant}\:\boldsymbol{{k}}\:{for}\:{which}\: \\ $$$$\sqrt{\left({x}^{\mathrm{2}} +{ax}\right)}\:−\:\sqrt{\left({x}^{\mathrm{2}} +{bx}\right)}\:<\:\boldsymbol{{k}}\:{for}\:{all}\:{x}>\mathrm{0}\: \\ $$$${is}\:? \\ $$

Question Number 33930 Answers: 1 Comments: 0

Three forces of magnitude 6N,2N and 3N act on the same point on the north,south and west directions respectively. find the magnitude and direction of the resultant force.

Question Number 33927 Answers: 0 Comments: 6

Question Number 33926 Answers: 0 Comments: 0

Question Number 33915 Answers: 2 Comments: 3

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) find Γ^((n)) (x) with n∈ N^★ 2) calculate Γ(n +(3/2)) for n integr.

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{with}\:{n}\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\:+\frac{\mathrm{3}}{\mathrm{2}}\right)\:{for}\:{n}\:{integr}. \\ $$

Question Number 33912 Answers: 0 Comments: 3

Question Number 33907 Answers: 0 Comments: 1

Question Number 33899 Answers: 2 Comments: 0

express ((4t^2 −28)/(t^4 +t^2 −6)) as a partial fraction.

$$\boldsymbol{\mathrm{express}}\:\frac{\mathrm{4}\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{28}}{\boldsymbol{\mathrm{t}}^{\mathrm{4}} +\boldsymbol{\mathrm{t}}^{\mathrm{2}} −\mathrm{6}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{partial}}\:\boldsymbol{\mathrm{fraction}}. \\ $$

Question Number 33897 Answers: 2 Comments: 0

let consider ψ(x)=((Γ^′ (x))/(Γ(x))) 1) prove that ∀ a>0 ∫_0 ^1 ψ(a+x)dx=ln(a) 2) prove that ∀ n∈ N^★ ∫_0 ^1 ψ(x)sin(2πnx)dx=−(π/2)

$${let}\:{consider}\:\psi\left({x}\right)=\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \psi\left({a}+{x}\right){dx}={ln}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\in\:{N}^{\bigstar} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\psi\left({x}\right){sin}\left(\mathrm{2}\pi{nx}\right){dx}=−\frac{\pi}{\mathrm{2}} \\ $$

Question Number 33896 Answers: 3 Comments: 0

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt 1) find Γ(x+1) interms of Γ(x) with x>0 2)calculate Γ(n) for n ∈ N^★ 3)calculate Γ((3/2)) .

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$

Question Number 33895 Answers: 3 Comments: 0

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) prove that Γ(x)Γ(1−x)= (π/(sin(πx))) 2) find the value of ∫_0 ^∞ e^(−x^2 ) dx .

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 33894 Answers: 0 Comments: 1

1)let f R→C 2π periodic even /f(x)=x ∀ x∈[0,π[ developp f at fourier serie 2) calculate Σ_(p=0) ^∞ (1/((2p+1)^2 )) .

$$\left.\mathrm{1}\right){let}\:{f}\:\:{R}\rightarrow{C}\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:/{f}\left({x}\right)={x}\: \\ $$$$\forall\:{x}\in\left[\mathrm{0},\pi\left[\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33892 Answers: 0 Comments: 1

prove that Σ_(n=1) ^∞ (H_n /n^2 ) =2 ξ(3) with ξ(x) =Σ_(n=1) ^∞ (1/n^x ) and x>1.

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:=\mathrm{2}\:\xi\left(\mathrm{3}\right)\:{with} \\ $$$$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{and}\:{x}>\mathrm{1}. \\ $$

Question Number 33891 Answers: 0 Comments: 1

let a∈C and ∣a∣<1 prove that the function f(x)= Σ_(n=0) ^(+∞) (a^n /(x+n)) is?developpable at point 1 and the radius is r=1.

$${let}\:{a}\in{C}\:{and}\:\mid{a}\mid<\mathrm{1}\:{prove}\:{that}\:{the}\:{function} \\ $$$${f}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{+\infty} \:\frac{{a}^{{n}} }{{x}+{n}}\:{is}?{developpable}\:{at}\:{point}\:\mathrm{1}\:{and} \\ $$$${the}\:{radius}\:{is}\:{r}=\mathrm{1}. \\ $$

Question Number 33890 Answers: 0 Comments: 0

find lim_(x→+∞) Σ_(n=1) ^∞ (1+(1/n))^n^2 (x^n /(n!)) .

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}^{\mathrm{2}} } \:\frac{{x}^{{n}} }{{n}!}\:. \\ $$

Question Number 33889 Answers: 0 Comments: 0

prove that ∫_0 ^1 (dx/(√(1−x^4 ))) = Σ_(n=0) ^∞ (C_(2n) ^n /(4^n (4n+1))) .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{C}_{\mathrm{2}{n}} ^{{n}} }{\mathrm{4}^{{n}} \left(\mathrm{4}{n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33888 Answers: 0 Comments: 0

developp at integr serie f(x)= ∫_0 ^(π/2) (dt/(√(1−x^2 sin^2 t))) . with ∣x∣<1 .

$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}}\:. \\ $$$${with}\:\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 33887 Answers: 0 Comments: 0

find the value of Σ_(n=0) ^∞ (((−1)^n )/(4n+1)) .

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{1}}\:. \\ $$

Question Number 33886 Answers: 0 Comments: 1

find the value of Σ_(n=0) ^∞ (((−1)^n )/(2n+3)).

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}{n}+\mathrm{3}}. \\ $$

Question Number 33885 Answers: 0 Comments: 1

developp at integr serie f(x)= ∫_0 ^x sin(t^2 )dt .

$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 33884 Answers: 0 Comments: 1

let F(x)= ∫_0 ^(π/2) ((arctan(xtant))/(tant)) dt find a simple form of f(x) . 2) find the value of ∫_0 ^(π/2) ((arctan(2tant))/(tant))dt .

$${let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{arctan}\left({xtant}\right)}{{tant}}\:{dt}\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{arctan}\left(\mathrm{2}{tant}\right)}{{tant}}{dt}\:. \\ $$

Question Number 33883 Answers: 0 Comments: 1

find a simple form of f(x)=∫_0 ^(π/2) ln(1+xsin^2 t)dt with ∣x∣<1.

$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} {t}\right){dt} \\ $$$${with}\:\mid{x}\mid<\mathrm{1}. \\ $$

Question Number 33867 Answers: 1 Comments: 0

Given A(t) is an area bounded between y = x^2 + tx and x−axis, 0 < t < 2 Find the propability we choose t so (1/(48)) ≤ A(t) ≤ (1/(16))

$$\mathrm{Given}\:{A}\left({t}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{between}\: \\ $$$${y}\:=\:{x}^{\mathrm{2}} \:+\:{tx}\:\mathrm{and}\:\mathrm{x}−\mathrm{axis},\:\:\mathrm{0}\:<\:{t}\:<\:\mathrm{2} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{propability}\:\mathrm{we}\:\mathrm{choose}\:{t} \\ $$$$\mathrm{so}\:\frac{\mathrm{1}}{\mathrm{48}}\:\leqslant\:{A}\left({t}\right)\:\leqslant\:\frac{\mathrm{1}}{\mathrm{16}} \\ $$

Question Number 33865 Answers: 0 Comments: 2

evaluate lim_(x→∞) π (((aπ)^x )/(x!))

$${evaluate} \\ $$$$\:{li}\underset{{x}\rightarrow\infty} {{m}}\:\:\:\pi\:\frac{\left({a}\pi\right)^{{x}} }{{x}!} \\ $$

Question Number 33880 Answers: 1 Comments: 1

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