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Question Number 33985    Answers: 0   Comments: 0

prove that (1/(1+cosx)) =2Σ_(n=1) ^∞ n(−1)^(n−1) cos(nx) for x≠kπ ,k∈ Z .

$${prove}\:{that}\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\mathrm{2}\sum_{{n}=\mathrm{1}} ^{\infty} {n}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} {cos}\left({nx}\right)\:{for} \\ $$$${x}\neq{k}\pi\:,{k}\in\:{Z}\:. \\ $$

Question Number 33984    Answers: 1   Comments: 1

calculate ∫_0 ^1 (1/x)ln(((1+x)/(1−x)))dx

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

Question Number 33983    Answers: 0   Comments: 1

find ∫_(1() ^∞ (1/x)ln(((x+1)/(x−1)))dx.

$${find}\:\int_{\mathrm{1}\left(\right.} ^{\infty} \frac{\mathrm{1}}{{x}}{ln}\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right){dx}. \\ $$

Question Number 33982    Answers: 0   Comments: 0

decompose inside R[x] the fraction F(x) = (1/(x^n (x+1)^2 )) with n integr .

$${decompose}\:{inside}\:{R}\left[{x}\right]\:{the}\:{fraction} \\ $$$${F}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}^{{n}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{with}\:{n}\:{integr}\:. \\ $$

Question Number 33981    Answers: 0   Comments: 0

let x∈[−1,1] andf_n (x)=sin(2narcsinx) 1)prove that f_n is odd and calculate f_n (0) and f_n (1) 2)solve inside [0^ 1] f_n (x)=0 3) prove that f_n is continue,derivable on[−1,1] and calculate f_n ^′ (x) 4) study the derivability of f_n at 1^− and (−1)^+ 5)calculate I_n = ∫_0 ^1 f_n (x)dx .

$${let}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right]\:{andf}_{{n}} \left({x}\right)={sin}\left(\mathrm{2}{narcsinx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}_{{n}} {is}\:{odd}\:{and}\:{calculate}\:{f}_{{n}} \left(\mathrm{0}\right)\:{and}\:{f}_{{n}} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){solve}\:{inside}\:\left[\bar {\mathrm{0}1}\right]\:\:{f}_{{n}} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{f}_{{n}} \:{is}\:{continue},{derivable}\:{on}\left[−\mathrm{1},\mathrm{1}\right]\:{and} \\ $$$${calculate}\:{f}_{{n}} ^{'} \left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{study}\:{the}\:{derivability}\:{of}\:{f}_{{n}} \:{at}\:\mathrm{1}^{−} \:\:{and}\:\left(−\mathrm{1}\right)^{+} \\ $$$$\left.\mathrm{5}\right){calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx}\:. \\ $$

Question Number 33980    Answers: 0   Comments: 0

1) let consider f(x)=∣cosx∣ π periodix developp f at fourier serie 2)find the valueof Σ_(n=1) ^∞ (((−1)^n )/(4n^2 −1)) 3)find the value of Σ_(n=1) ^∞ (1/((4n^2 −1)^2 )) .

$$\left.\mathrm{1}\right)\:{let}\:{consider}\:{f}\left({x}\right)=\mid{cosx}\mid\:\pi\:{periodix} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{valueof}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33979    Answers: 0   Comments: 1

we give for t>0 ∫_0 ^∞ ((sinx)/x) e^(−tx) dx =(π/2) −arctant use this result to find the value of ∫_0 ^∞ (((1−e^(−x) )sinx)/x^2 )dx .

$${we}\:{give}\:{for}\:{t}>\mathrm{0}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{x}}\:{e}^{−{tx}} {dx}\:=\frac{\pi}{\mathrm{2}}\:−{arctant} \\ $$$${use}\:{this}\:{result}\:{to}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 33978    Answers: 1   Comments: 2

let f(t) = ∫_0 ^∞ ((sin(x^2 )e^(−tx^2 ) )/x^2 ) dx with t>0 find a simple form of f^′ (t) .

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right){e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }\:{dx}\:\:\:\:\:\:\:{with}\:{t}>\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}^{'} \left({t}\right)\:. \\ $$

Question Number 33977    Answers: 1   Comments: 0

find a polynome solution of the diifferencial equation y^(′′) +y =x^(12) .

$${find}\:{a}\:{polynome}\:{solution}\:{of}\:{the}\:{diifferencial} \\ $$$${equation}\:{y}^{''} \:+{y}\:={x}^{\mathrm{12}} \:\:. \\ $$

Question Number 33969    Answers: 2   Comments: 0

if tanA+tanB=P and AtanB=q express the value of cos(2A+3B) in terms of p and q.

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{tanA}}+\boldsymbol{\mathrm{tanB}}=\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{AtanB}}=\boldsymbol{\mathrm{q}} \\ $$$$\boldsymbol{\mathrm{express}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{cos}}\left(\mathrm{2}\boldsymbol{\mathrm{A}}+\mathrm{3}\boldsymbol{\mathrm{B}}\right)\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{p}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{q}}. \\ $$

Question Number 33944    Answers: 1   Comments: 0

If the equation (p^2 −4)(p^2 −9)x^3 +[((p−2)/2)]x^2 +(p−4)(p−3)(p−2)x+{2p−1}=0. is satisfied by all values of x in (0,3] then sum of all possible integral values of ′p′ is ? {.} = fractional part function. [.]= greatest integer function.

$$\boldsymbol{\mathrm{I}}\mathrm{f}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\left(\mathrm{p}^{\mathrm{2}} −\mathrm{4}\right)\left(\mathrm{p}^{\mathrm{2}} −\mathrm{9}\right){x}^{\mathrm{3}} +\left[\frac{\mathrm{p}−\mathrm{2}}{\mathrm{2}}\right]{x}^{\mathrm{2}} +\left(\mathrm{p}−\mathrm{4}\right)\left(\mathrm{p}−\mathrm{3}\right)\left(\mathrm{p}−\mathrm{2}\right){x}+\left\{\mathrm{2p}−\mathrm{1}\right\}=\mathrm{0}. \\ $$$$\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{in}\:\left(\mathrm{0},\mathrm{3}\right]\:{then} \\ $$$${sum}\:{of}\:{all}\:{possible}\:{integral}\:{values}\:{of} \\ $$$$'{p}'\:{is}\:? \\ $$$$\left\{.\right\}\:=\:{fractional}\:{part}\:{function}. \\ $$$$\left[.\right]=\:{greatest}\:{integer}\:{function}. \\ $$

Question Number 33942    Answers: 1   Comments: 1

Question Number 33941    Answers: 0   Comments: 0

9

$$\mathrm{9} \\ $$

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.

$$\boldsymbol{\mathrm{Three}}\:\boldsymbol{\mathrm{forces}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{magnitude}}\:\mathrm{6}\boldsymbol{\mathrm{N}},\mathrm{2}\boldsymbol{\mathrm{N}} \\ $$$$\boldsymbol{\mathrm{and}}\:\mathrm{3}\boldsymbol{\mathrm{N}}\:\boldsymbol{\mathrm{act}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{point}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{north}},\boldsymbol{\mathrm{south}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{west}}\:\boldsymbol{\mathrm{directions}}\:\boldsymbol{\mathrm{respectively}}. \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{magnitude}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{direction}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{resultant}}\:\boldsymbol{\mathrm{force}}. \\ $$$$ \\ $$

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.

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

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