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Relation and FunctionsQuestion and Answers: Page 8

Question Number 146550    Answers: 0   Comments: 0

f(x)=x^3 arctan((2/x)) 1)calculate f^((n)) (x) 2)developp f at integr srie at x_0 =1

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} \:\mathrm{arctan}\left(\frac{\mathrm{2}}{\mathrm{x}}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculate}\:\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right) \\ $$$$\left.\mathrm{2}\right)\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{srie}\:\mathrm{at}\:\mathrm{x}_{\mathrm{0}} =\mathrm{1} \\ $$

Question Number 146549    Answers: 2   Comments: 0

find lim_(x→0) ((cos(x−sinx)+1−cos(x^2 ))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{cos}\left(\mathrm{x}−\mathrm{sinx}\right)+\mathrm{1}−\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 146548    Answers: 2   Comments: 0

let f(x)=cos(αx) developp f at fourier serie (α real)

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{cos}\left(\alpha\mathrm{x}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie}\:\:\left(\alpha\:\mathrm{real}\right) \\ $$

Question Number 146547    Answers: 3   Comments: 0

calculate ∫_(∣z∣=5) ((2−z^2 )/((z^2 +9)(z−i)^2 ))dz

$$\mathrm{calculate}\:\int_{\mid\mathrm{z}\mid=\mathrm{5}} \:\:\:\frac{\mathrm{2}−\mathrm{z}^{\mathrm{2}} }{\left(\mathrm{z}^{\mathrm{2}} +\mathrm{9}\right)\left(\mathrm{z}−\mathrm{i}\right)^{\mathrm{2}} }\mathrm{dz} \\ $$

Question Number 146546    Answers: 2   Comments: 0

find ∫_(∣z−1∣=3) ((cos(πz))/((z−2)(z^2 +4)))dz

$$\mathrm{find}\:\int_{\mid\mathrm{z}−\mathrm{1}\mid=\mathrm{3}} \:\:\frac{\mathrm{cos}\left(\pi\mathrm{z}\right)}{\left(\mathrm{z}−\mathrm{2}\right)\left(\mathrm{z}^{\mathrm{2}} +\mathrm{4}\right)}\mathrm{dz} \\ $$

Question Number 146524    Answers: 0   Comments: 0

find ∫_0 ^∞ ((xsinx)/((x^4 +1)^3 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{xsinx}}{\left(\mathrm{x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dx} \\ $$

Question Number 146497    Answers: 1   Comments: 0

f(x)=(2/x)∫_0 ^x (t^2 /( (√(1+t^2 ))))dt calculate lim_(x→0) f(x)

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\mathrm{x}}\int_{\mathrm{0}} ^{\mathrm{x}} \:\:\frac{\mathrm{t}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\mathrm{dt}\:\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \mathrm{f}\left(\mathrm{x}\right) \\ $$

Question Number 146363    Answers: 1   Comments: 0

find ∫_(−i) ^(1+i) (x^2 −iy)dz along y=x^3

$$\mathrm{find}\:\int_{−\mathrm{i}} ^{\mathrm{1}+\mathrm{i}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{iy}\right)\mathrm{dz}\:\:\mathrm{along}\:\:\mathrm{y}=\mathrm{x}^{\mathrm{3}} \\ $$

Question Number 146260    Answers: 1   Comments: 0

solve y^(′′) −2y^′ +y =e^(−x) sinx

$$\mathrm{solve}\:\mathrm{y}^{''} −\mathrm{2y}^{'} \:+\mathrm{y}\:=\mathrm{e}^{−\mathrm{x}} \mathrm{sinx} \\ $$

Question Number 146365    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((arctan(x^2 ))/(x^2 +4))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\mathrm{dx} \\ $$

Question Number 146196    Answers: 2   Comments: 0

calculate ∫_0 ^∞ (dx/((2x+1)^4 (x+3)^5 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{2x}+\mathrm{1}\right)^{\mathrm{4}} \left(\mathrm{x}+\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 146197    Answers: 2   Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)^3 ))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 146194    Answers: 1   Comments: 0

calculate Σ_(n=1) ^∞ (1/(n^3 5^n ))

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \mathrm{5}^{\mathrm{n}} } \\ $$

Question Number 146193    Answers: 2   Comments: 0

solve y^(′′) −y^′ + y=xe^(−x)

$$\mathrm{solve}\:\mathrm{y}^{''} \:−\mathrm{y}^{'} \:+\:\mathrm{y}=\mathrm{xe}^{−\mathrm{x}} \\ $$

Question Number 146087    Answers: 1   Comments: 0

1)find U_n =∫_0 ^1 x^n e^(−2x) dx 2)nature of Σ U_n ?

$$\left.\mathrm{1}\right)\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} ? \\ $$

Question Number 146085    Answers: 0   Comments: 1

f(x,y)=x−(√(x+2y)) 1)condition on x and y to have f symetric 2) find (∂f/∂x) ,(∂f/∂y) ,(∂^2 f/(∂x∂y)) ,(∂^2 f/(∂y∂x)) 3) find (∂^2 f/∂^2 x) and (∂^2 f/∂^2 y)

$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}−\sqrt{\mathrm{x}+\mathrm{2y}} \\ $$$$\left.\mathrm{1}\right)\mathrm{condition}\:\mathrm{on}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{to}\:\mathrm{have}\:\mathrm{f}\:\mathrm{symetric} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\frac{\partial\mathrm{f}}{\partial\mathrm{x}}\:,\frac{\partial\mathrm{f}}{\partial\mathrm{y}}\:,\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial\mathrm{x}\partial\mathrm{y}}\:,\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial\mathrm{y}\partial\mathrm{x}} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial^{\mathrm{2}} \mathrm{x}}\:\mathrm{and}\:\frac{\partial^{\mathrm{2}} \mathrm{f}}{\partial^{\mathrm{2}} \mathrm{y}} \\ $$

Question Number 146083    Answers: 1   Comments: 0

F(x)=x^n −e^(inα) 1) roots of F(x)? 2) factorize F(x) inside C[x]

$$\mathrm{F}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} \:−\mathrm{e}^{\mathrm{in}\alpha} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{F}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$

Question Number 146082    Answers: 0   Comments: 0

p(x)=(x^2 −x+1)^n −(x^2 +x+1)^n 1) roots of p(x)? 2) factorize p(x) inside C[x]

$$\mathrm{p}\left(\mathrm{x}\right)=\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{p}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{p}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$

Question Number 146072    Answers: 0   Comments: 0

Let F_n =2^2^n +1 the fermat number Prove that F_n is prime ⇔ 3^((F_n −1)/2) ≡1[F_n ]

$${Let}\:{F}_{{n}} =\mathrm{2}^{\mathrm{2}^{{n}} } +\mathrm{1}\:{the}\:{fermat}\:{number} \\ $$$${Prove}\:{that} \\ $$$$\:{F}_{{n}} \:{is}\:{prime}\:\Leftrightarrow\:\mathrm{3}^{\frac{{F}_{{n}} −\mathrm{1}}{\mathrm{2}}} \equiv\mathrm{1}\left[{F}_{{n}} \right] \\ $$

Question Number 146061    Answers: 0   Comments: 0

Question Number 145960    Answers: 2   Comments: 0

Question Number 145941    Answers: 1   Comments: 0

find ∫_0 ^∞ e^(−3x) log(1+x^3 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{3}{x}} {log}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx} \\ $$

Question Number 145940    Answers: 0   Comments: 0

find ∫_0 ^1 e^(−x) log(1−x^4 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−{x}} {log}\left(\mathrm{1}−{x}^{\mathrm{4}} \right){dx} \\ $$

Question Number 145939    Answers: 0   Comments: 0

Ψ(x)=ch(sinx) developp Ψ at fourier serie

$$\Psi\left({x}\right)={ch}\left({sinx}\right) \\ $$$${developp}\:\Psi\:{at}\:{fourier}\:{serie} \\ $$

Question Number 145938    Answers: 1   Comments: 0

g(x)=cos(arctanx) if g(x)=Σ a_n x^n determine the sequence a_n

$${g}\left({x}\right)={cos}\left({arctanx}\right) \\ $$$${if}\:{g}\left({x}\right)=\Sigma\:{a}_{{n}} {x}^{{n}} \:{determine}\:{the} \\ $$$${sequence}\:{a}_{{n}} \\ $$

Question Number 145936    Answers: 0   Comments: 0

g(x)=arctan(cosx) developp f at fourier serie

$${g}\left({x}\right)={arctan}\left({cosx}\right) \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$

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