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Question Number 141691    Answers: 1   Comments: 0

....Calculus(I).... 𝛗:=∫_(1/(2 )) ^( 1) (1/(x^2 (1+x^4 )^(3/4) ))dx=???

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{Calculus}\left({I}\right).... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\frac{\mathrm{1}}{\mathrm{2}\:}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\frac{\mathrm{3}}{\mathrm{4}}} }{dx}=??? \\ $$

Question Number 141685    Answers: 1   Comments: 0

......nice ... ... ... calculus..... If lim_(xβ†’0) ((tan(x))/x) = 1 , prove that: lim(1/x)((1/x)βˆ’(1/(tan(x))))=(1/3)

$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:......{nice}\:...\:...\:...\:{calculus}..... \\ $$$$\:\:\mathrm{I}{f}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{tan}\left({x}\right)}{{x}}\:=\:\mathrm{1}\:,\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:{lim}\frac{\mathrm{1}}{{x}}\left(\frac{\mathrm{1}}{{x}}βˆ’\frac{\mathrm{1}}{{tan}\left({x}\right)}\right)=\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 141681    Answers: 1   Comments: 0

On the Argand Diagram, the variable point Z represents a complex number z. Find the equation of the locus of a point Z which moves such that ∣((zβˆ’1)/(z+2))∣=2

$$\mathrm{On}\:\mathrm{the}\:\mathrm{Argand}\:\mathrm{Diagram},\:\mathrm{the}\:\mathrm{variable}\:\mathrm{point} \\ $$$$\mathrm{Z}\:\mathrm{represents}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}\:{z}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{a}\:\mathrm{point} \\ $$$$\mathrm{Z}\:\mathrm{which}\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mid\frac{{z}βˆ’\mathrm{1}}{{z}+\mathrm{2}}\mid=\mathrm{2} \\ $$

Question Number 143167    Answers: 2   Comments: 0

∫arctan((√((√x)+1)))dx=??? proposeβ€² par Rodrigue

$$\int\boldsymbol{{arctan}}\left(\sqrt{\sqrt{\boldsymbol{{x}}}+\mathrm{1}}\right)\boldsymbol{{dx}}=??? \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{propose}}'\:\boldsymbol{{par}}\:\boldsymbol{{Rodrigue}} \\ $$

Question Number 141672    Answers: 1   Comments: 0

Solve the equation x^4 βˆ’2x^3 βˆ’5x^2 +10xβˆ’3=0

$$\:\:{Solve}\:{the}\:{equation}\: \\ $$$$\:\:{x}^{\mathrm{4}} βˆ’\mathrm{2}{x}^{\mathrm{3}} βˆ’\mathrm{5}{x}^{\mathrm{2}} +\mathrm{10}{x}βˆ’\mathrm{3}=\mathrm{0} \\ $$

Question Number 141668    Answers: 1   Comments: 0

.......Advanced ...β˜… ...β˜… ... Calculus....... if Ξ© =Ξ£_(n=2) ^∞ (((βˆ’1)^n ΞΆ(n))/2^n ) then prove that :: (1/2) = e^(Ξ©βˆ’1) proof :: method (1): ψ (1+x )= βˆ’Ξ³+Ξ£_(n=2) ^∞ (βˆ’1)^n ΞΆ(n)x^(nβˆ’1) ( Maclaurin series for ψ(x+1) ) x:=(1/2) β‡’ ψ ((3/2) )=βˆ’Ξ³ + 2Ξ£_(n=2) ^∞ (((βˆ’1)^n ΞΆ(n))/2^n ) (βˆ— ) we know that :: ψ(1+x)=(1/x)+ψ(x) ( βˆ— ) ⇛ ψ ((3/2))=2+ψ((1/2))=βˆ’Ξ³+2Ξ£_(n=2) ^∞ (((βˆ’1)^n ΞΆ(n))/2^n ) (βˆ—) ⇛ 2βˆ’Ξ³βˆ’ln(4)=βˆ’Ξ³+2Ξ£_(n=2) ^∞ (((βˆ’1)^n ΞΆ(n))/2^n ) ln((e/2))= Ξ£_(n=2) ^∞ (((βˆ’1)^n ΞΆ(n))/2^n ) =Ξ© (1/2) = e^(Ξ© βˆ’1) ....βœ“ ...m.n.july.1970...

$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:.......{Advanced}\:...\bigstar\:...\bigstar\:...\:{Calculus}....... \\ $$$$\:\:\:\:\:\:\:\:\:{if}\:\:\:\:\Omega\:=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} }\:{then}\:{prove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{that}\:::\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:=\:{e}^{\Omegaβˆ’\mathrm{1}} \:\: \\ $$$$\:\:\:\:{proof}\::: \\ $$$$\:\:\:\:{method}\:\left(\mathrm{1}\right): \\ $$$$\:\:\:\:\:\psi\:\left(\mathrm{1}+{x}\:\right)=\:βˆ’\gamma+\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right){x}^{{n}βˆ’\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\left(\:{Maclaurin}\:{series}\:{for}\:\psi\left({x}+\mathrm{1}\right)\:\right) \\ $$$$\:\:\:\:{x}:=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:\psi\:\left(\frac{\mathrm{3}}{\mathrm{2}}\:\right)=βˆ’\gamma\:+\:\mathrm{2}\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} }\:\:\left(\ast\:\right) \\ $$$$\:\:\:\:{we}\:{know}\:{that}\:::\:\psi\left(\mathrm{1}+{x}\right)=\frac{\mathrm{1}}{{x}}+\psi\left({x}\right) \\ $$$$\:\:\:\:\:\:\left(\:\ast\:\right)\:\:\Rrightarrow\:\psi\:\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\mathrm{2}+\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=βˆ’\gamma+\mathrm{2}\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} } \\ $$$$\:\:\:\:\:\:\:\left(\ast\right)\:\:\:\:\:\Rrightarrow\:\:\:\:\:\:\:\:\:\mathrm{2}βˆ’\gammaβˆ’{ln}\left(\mathrm{4}\right)=βˆ’\gamma+\mathrm{2}\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ln}\left(\frac{{e}}{\mathrm{2}}\right)=\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} }\:=\Omega \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:=\:{e}^{\Omega\:βˆ’\mathrm{1}} \:\:\:....\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 141665    Answers: 0   Comments: 1

Question Number 141663    Answers: 0   Comments: 1

_( βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’) x^2 (xβˆ’12)(xβˆ’15)=k(xβˆ’16) find x in terms of k (>0). ^(βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’)

$$\:\underset{\:\:βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’} {\:} \\ $$$$\:\:{x}^{\mathrm{2}} \left({x}βˆ’\mathrm{12}\right)\left({x}βˆ’\mathrm{15}\right)={k}\left({x}βˆ’\mathrm{16}\right) \\ $$$$\:\:\:\:{find}\:{x}\:{in}\:{terms}\:{of}\:{k}\:\left(>\mathrm{0}\right). \\ $$$$\:\:\overset{βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’} {\:} \\ $$

Question Number 141658    Answers: 1   Comments: 0

Question Number 141654    Answers: 0   Comments: 2

Solve for real numbers 5βˆ™(((1βˆ’z))^(1/5) + ((1+z))^(1/5) = 2+4βˆ™(((1βˆ’z))^(1/4) + ((1+z))^(1/4) )

$${Solve}\:{for}\:{real}\:{numbers} \\ $$$$\mathrm{5}\centerdot\left(\sqrt[{\mathrm{5}}]{\mathrm{1}βˆ’{z}}\:+\:\sqrt[{\mathrm{5}}]{\mathrm{1}+{z}}\:=\:\mathrm{2}+\mathrm{4}\centerdot\left(\sqrt[{\mathrm{4}}]{\mathrm{1}βˆ’{z}}\:+\:\sqrt[{\mathrm{4}}]{\mathrm{1}+{z}}\right)\right. \\ $$

Question Number 141653    Answers: 0   Comments: 0

what is condition to have log( I +A)=Σ a_n A^n and determine the sequence (a_n ) A ∈ M_n (C)

$${what}\:{is}\:{condition}\:{to}\:{have} \\ $$$${log}\left(\:{I}\:+{A}\right)=\Sigma\:{a}_{{n}} {A}^{{n}} \\ $$$${and}\:{determine}\:{the}\:{sequence}\:\left({a}_{{n}} \right) \\ $$$${A}\:\in\:{M}_{{n}} \left({C}\right) \\ $$

Question Number 141652    Answers: 0   Comments: 0

A = (((1 2)),((βˆ’1 1)) ) find e^(A ) and e^(tA) find ch(A) and sh(A)

$${A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\\{βˆ’\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${find}\:{e}^{{A}\:} \:{and}\:{e}^{{tA}} \\ $$$${find}\:{ch}\left({A}\right)\:{and}\:{sh}\left({A}\right) \\ $$

Question Number 141651    Answers: 0   Comments: 2

Question Number 141649    Answers: 1   Comments: 0

.......advanced calculus...... prove thatβˆ’:: Ο†:=∫_0 ^( ∞) ((cos(2Ο€x^2 ))/(cosh^2 (Ο€x)))dx=(1/4) βœ“

$$\:\:\:\:\:\:\:\:\:.......{advanced}\:\:{calculus}...... \\ $$$$\:\:\:\:{prove}\:\:{that}βˆ’:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\phi:=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{cos}\left(\mathrm{2}\pi{x}^{\mathrm{2}} \right)}{{cosh}^{\mathrm{2}} \left(\pi{x}\right)}{dx}=\frac{\mathrm{1}}{\mathrm{4}}\:\:\checkmark \\ $$

Question Number 141937    Answers: 0   Comments: 2

Write and graph the equation of the graph of y=sin(Ο€x) It is stretched up by a factor of 5 and shifted (1/2) unit to the right Help me please

$${Write}\:{and}\:{graph}\:{the}\:{equation}\:{of}\:{the}\:{graph}\:{of}\:{y}={sin}\left(\pi{x}\right) \\ $$$${It}\:{is}\:{stretched}\:{up}\:{by}\:{a}\:{factor}\:{of}\:\mathrm{5}\:{and}\:{shifted}\:\frac{\mathrm{1}}{\mathrm{2}}\:{unit}\:{to}\:{the}\:{right} \\ $$$${Help}\:{me}\:{please} \\ $$$$ \\ $$

Question Number 141645    Answers: 0   Comments: 0

Question Number 141643    Answers: 1   Comments: 0

Question Number 141642    Answers: 0   Comments: 0

Question Number 141640    Answers: 1   Comments: 0

Question Number 141635    Answers: 0   Comments: 0

solve the differential equation (PDE), z((βˆ‚z/βˆ‚x)βˆ’(βˆ‚z/βˆ‚y))=z^2 +(x+y)^2 .

$${solve}\:{the}\:{differential}\:{equation}\:\left({PDE}\right), \\ $$$${z}\left(\frac{\partial{z}}{\partial{x}}βˆ’\frac{\partial{z}}{\partial{y}}\right)={z}^{\mathrm{2}} +\left({x}+{y}\right)^{\mathrm{2}} . \\ $$

Question Number 141633    Answers: 0   Comments: 0

1<a<b ,prove that : b^n = Ξ£_(k=0) ^n (βˆ’1)^k C_n ^k a^((ln(Ξ£_(p=0) ^(nβˆ’k) C_(nβˆ’k) ^p a^(nβˆ’p) b^p ))/(ln(a)))

$$\mathrm{1}<\mathrm{a}<\mathrm{b}\:,\mathrm{prove}\:\mathrm{that}\:: \\ $$$${b}^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(βˆ’\mathrm{1}\right)^{{k}} \mathrm{C}_{{n}} ^{{k}} \:{a}^{\frac{{ln}\left(\underset{{p}=\mathrm{0}} {\overset{{n}βˆ’{k}} {\sum}}\mathrm{C}_{{n}βˆ’{k}} ^{{p}} {a}^{{n}βˆ’{p}} {b}^{{p}} \right)}{{ln}\left({a}\right)}} \\ $$

Question Number 141632    Answers: 0   Comments: 0

Let f(x)=((sin(x))/x) , prove that : Ξ£_(n=0) ^∞ [ f(nΟ€+Ξ±)+f(nΟ€βˆ’Ξ±) ]= 1+f(Ξ±)

$$\mathrm{Let}\:{f}\left({x}\right)=\frac{{sin}\left({x}\right)}{{x}}\:,\:\mathrm{prove}\:\mathrm{that}\:: \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\:{f}\left({n}\pi+\alpha\right)+{f}\left({n}\piβˆ’\alpha\right)\:\right]=\:\mathrm{1}+{f}\left(\alpha\right) \\ $$

Question Number 141669    Answers: 1   Comments: 0

Question Number 141628    Answers: 0   Comments: 3

Question Number 141627    Answers: 2   Comments: 0

Question Number 141623    Answers: 1   Comments: 0

Σ_(n=0) ^∞ (∫_0 ^1 (x^n /(1+x))dx)^2 =ln 2

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\right)^{\mathrm{2}} ={ln}\:\mathrm{2} \\ $$

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