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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.

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

$$\mathrm{9} \\ $$

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.

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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