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Question Number 92869    Answers: 0   Comments: 7

∫((−t^3 +2t−t+1)/(t(t^2 +1)))dt

$$\int\frac{−{t}^{\mathrm{3}} +\mathrm{2}{t}−{t}+\mathrm{1}}{{t}\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{dt} \\ $$

Question Number 92862    Answers: 0   Comments: 0

Exercise (D):y=x−1 directed by u^→ (1;1). his normal vector is v^→ (2;−2). 1) Determinate the equation of (Δ) such as t_v^→ =S_((D)) °S_((Δ)) 2)Determinate the nature and caracteristics of this application h=S_((D)) °t_v^→

$$\mathrm{Exercise} \\ $$$$\: \\ $$$$\left(\mathrm{D}\right):\mathrm{y}=\mathrm{x}−\mathrm{1}\:\:\mathrm{directed}\:\mathrm{by}\:\:\:\overset{\rightarrow} {\mathrm{u}}\left(\mathrm{1};\mathrm{1}\right). \\ $$$$\mathrm{his}\:\mathrm{normal}\:\mathrm{vector}\:\mathrm{is}\:\overset{\rightarrow} {\mathrm{v}}\left(\mathrm{2};−\mathrm{2}\right). \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Determinate}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\:\left(\Delta\right)\: \\ $$$$\mathrm{such}\:\mathrm{as}\:\mathrm{t}_{\overset{\rightarrow} {\mathrm{v}}} =\mathrm{S}_{\left(\mathrm{D}\right)} °\mathrm{S}_{\left(\Delta\right)} \\ $$$$\left.\mathrm{2}\right)\mathrm{Determinate}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{and}\:\mathrm{caracteristics} \\ $$$$\mathrm{of}\:\mathrm{this}\:\mathrm{application}\:\mathrm{h}=\mathrm{S}_{\left(\mathrm{D}\right)} °\mathrm{t}_{\overset{\rightarrow} {\mathrm{v}}} \\ $$$$ \\ $$

Question Number 92860    Answers: 0   Comments: 2

(√(1−x^2 )) sin^(−1) (x) dy + y dx = 0

$$\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{dy}\:+\:\mathrm{y}\:\mathrm{dx}\:=\:\mathrm{0}\: \\ $$

Question Number 92852    Answers: 3   Comments: 1

y′′+2y′+y = x^2 e^(−x) cos x what is particular solution

$$\mathrm{y}''+\mathrm{2y}'+\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \mathrm{e}^{−\mathrm{x}} \mathrm{cos}\:\mathrm{x} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{particular}\:\mathrm{solution} \\ $$$$ \\ $$

Question Number 92839    Answers: 1   Comments: 0

let a is complex number such that a^(10) + a^5 +1 = 0. find a^(2005) + (1/a^(2005) ) ?

$$\mathrm{let}\:\mathrm{a}\:\mathrm{is}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{such}\: \\ $$$$\mathrm{that}\:\mathrm{a}^{\mathrm{10}} \:+\:\mathrm{a}^{\mathrm{5}} \:+\mathrm{1}\:=\:\mathrm{0}. \\ $$$$\mathrm{find}\:\mathrm{a}^{\mathrm{2005}} \:+\:\frac{\mathrm{1}}{\mathrm{a}^{\mathrm{2005}} }\:? \\ $$

Question Number 92838    Answers: 0   Comments: 4

(x+y)(dy/dx) = x^2 +xy+x+1

$$\left(\mathrm{x}+\mathrm{y}\right)\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{xy}+\mathrm{x}+\mathrm{1} \\ $$

Question Number 92835    Answers: 0   Comments: 3

Question Number 92831    Answers: 1   Comments: 5

Question Number 92820    Answers: 1   Comments: 0

a convergent geometric sequence with first term a is such that the sum of the terms after the n^(th) term is three times the n^(th) term, find the common ratio and show that its sum to infinity is 4a.

$${a}\:{convergent}\:{geometric}\:{sequence}\:{with} \\ $$$${first}\:{term}\:{a}\:{is}\:{such}\:{that}\:{the}\:{sum}\:{of} \\ $$$${the}\:{terms}\:{after}\:{the}\:{n}^{{th}} \:{term}\:{is} \\ $$$${three}\:{times}\:{the}\:{n}^{{th}} \:{term},\:{find}\:{the} \\ $$$${common}\:{ratio}\:{and}\:{show}\:{that}\:{its}\: \\ $$$${sum}\:{to}\:{infinity}\:{is}\:\mathrm{4}{a}. \\ $$

Question Number 92808    Answers: 0   Comments: 0

Can you prove it: t=Φ

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{prove}\:{it}: \\ $$$${t}=\Phi \\ $$

Question Number 92807    Answers: 2   Comments: 1

Question Number 92804    Answers: 1   Comments: 12

Solve the following differential equations: (i). e^(x−y) dx +e^(y−x) dy=0 (ii). (dy/dx) = (√(y−x)) (iii). (dy/dx)= ((3xy+y^2 )/(3x^2 ))

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{differential}\:\mathrm{equations}: \\ $$$$\:\left(\mathrm{i}\right).\:\mathrm{e}^{\mathrm{x}−\mathrm{y}} \:\mathrm{dx}\:+\mathrm{e}^{\mathrm{y}−\mathrm{x}} \:\mathrm{dy}=\mathrm{0} \\ $$$$\:\:\left(\mathrm{ii}\right).\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\sqrt{\mathrm{y}−\mathrm{x}} \\ $$$$\:\left(\mathrm{iii}\right).\:\frac{\mathrm{dy}}{\mathrm{dx}}=\:\frac{\mathrm{3xy}+\mathrm{y}^{\mathrm{2}} }{\mathrm{3x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 92805    Answers: 1   Comments: 0

Evaluate: ∫_R ∫ ((xy)/(√(1−y^2 ))) dx dy where the region of integration is the positive quadrant of the circle x^2 +y^2 =1.

$$\:\mathrm{Evaluate}: \\ $$$$\:\int_{\boldsymbol{\mathrm{R}}} \int\:\frac{\boldsymbol{\mathrm{xy}}}{\sqrt{\mathrm{1}−\boldsymbol{\mathrm{y}}^{\mathrm{2}} }}\:\boldsymbol{\mathrm{dx}}\:\boldsymbol{\mathrm{dy}}\:\mathrm{where}\:\mathrm{the}\:\mathrm{region}\:\mathrm{of}\:\mathrm{integration}\:\mathrm{is}\:\mathrm{the} \\ $$$$\:\mathrm{positive}\:\mathrm{quadrant}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{1}. \\ $$$$ \\ $$

Question Number 92801    Answers: 0   Comments: 0

Integrate following : (i).∫ (( dx)/(sin x( 3+2cos x))) (ii).∫(√((sin(x−α))/(sin(x+α)))) dx

$$\boldsymbol{\mathrm{Integrate}}\:\boldsymbol{\mathrm{following}}\:: \\ $$$$\:\:\left(\boldsymbol{\mathrm{i}}\right).\int\:\frac{\:\:\mathrm{dx}}{\mathrm{sin}\:\mathrm{x}\left(\:\mathrm{3}+\mathrm{2cos}\:\mathrm{x}\right)} \\ $$$$\:\:\left(\boldsymbol{\mathrm{ii}}\right).\int\sqrt{\frac{\mathrm{sin}\left(\mathrm{x}−\alpha\right)}{\mathrm{sin}\left(\mathrm{x}+\alpha\right)}}\:\:\mathrm{dx}\: \\ $$$$ \\ $$

Question Number 92799    Answers: 0   Comments: 2

Show that the function x→x^3 is of Riemann within the interval [−1,2] then calculate ∫_(−1) ^2 x^2 dx

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{x}\rightarrow\mathrm{x}^{\mathrm{3}} \:\mathrm{is} \\ $$$$\mathrm{of}\:\mathrm{Riemann}\:\mathrm{within}\:\mathrm{the}\:\mathrm{interval}\:\left[−\mathrm{1},\mathrm{2}\right] \\ $$$$\mathrm{then}\:\mathrm{calculate}\:\int_{−\mathrm{1}} ^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \mathrm{dx} \\ $$

Question Number 92798    Answers: 0   Comments: 0

Find the value of 𝛉 in the Mean Value Theorem f(x+h) = f(x) +h f^( ′) (x+θh) if f(x)= (1/x) .

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\theta}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Mean}\:\mathrm{Value} \\ $$$$\:\:\mathrm{Theorem}\: \\ $$$$\:\:\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)\:+\mathrm{h}\:\mathrm{f}^{\:'} \left(\mathrm{x}+\theta\mathrm{h}\right)\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\:\frac{\mathrm{1}}{\mathrm{x}}\:. \\ $$

Question Number 92795    Answers: 0   Comments: 0

Define Clairaut′s equation and solve y= px +(√(a^2 p^2 +b^2 ))

$$\boldsymbol{\mathrm{Define}}\:\boldsymbol{\mathrm{Clairaut}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{equation}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{solve}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{px}}\:+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} \boldsymbol{\mathrm{p}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} } \\ $$

Question Number 92790    Answers: 0   Comments: 0

∫_0 ^1 ((π/4)−tan^(−1) (x))(dx/(1−x^2 ))

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\pi}{\mathrm{4}}−{tan}^{−\mathrm{1}} \left({x}\right)\right)\frac{{dx}}{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$

Question Number 92788    Answers: 0   Comments: 0

Question Number 92785    Answers: 0   Comments: 2

a) Find E(x^x ) then E(x^x^x ) for x∈]0,1[ b) find lim_(x→0) E(x^x^x )

$$\left.\mathrm{a}\left.\right)\:\mathrm{Find}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}} \right)\:\mathrm{then}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \right)\:\mathrm{for}\:\mathrm{x}\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \right) \\ $$

Question Number 92782    Answers: 0   Comments: 0

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Question Number 92781    Answers: 1   Comments: 1

tan^(−1) (x) = sin^(−1) (((2x)/(x^2 +1)))

$$\mathrm{tan}^{−\mathrm{1}} \left({x}\right)\:=\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\right)\: \\ $$

Question Number 92778    Answers: 0   Comments: 1

Question Number 92772    Answers: 0   Comments: 1

Question Number 92771    Answers: 0   Comments: 0

Question Number 92769    Answers: 0   Comments: 0

calculate Σ_(k=0) ^(n−1) C_n ^k cos^2 (((kπ)/n)) (n≥2)

$${calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{C}_{{n}} ^{{k}} \:{cos}^{\mathrm{2}} \left(\frac{{k}\pi}{{n}}\right)\:\:\:\:\:\left({n}\geqslant\mathrm{2}\right) \\ $$

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