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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)

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

Question Number 92768 Answers: 0 Comments: 0

calculate A_n = Σ_(k=0) ^n C_n ^k cos^3 (kx) and Σ_(k=0) ^n C_n ^k sin^3 (kx)

$${calculate}\:{A}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{\mathrm{3}} \left({kx}\right)\:\:{and}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{sin}^{\mathrm{3}} \left({kx}\right) \\ $$

Question Number 92763 Answers: 0 Comments: 1

study the sequence u_(n+1) =(√(u_n ^2 +(1/n))) and u_1 =1

$${study}\:{the}\:{sequence}\:{u}_{{n}+\mathrm{1}} =\sqrt{{u}_{{n}} ^{\mathrm{2}} +\frac{\mathrm{1}}{{n}}} \\ $$$${and}\:{u}_{\mathrm{1}} =\mathrm{1} \\ $$

Question Number 92767 Answers: 0 Comments: 0

let U_n =∫_0 ^1 e^(−nx) cosx dx 1) determine lim_(n→+∞) n^2 U_n 2)calculate Σ_(n=0) ^∞ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−{nx}} {cosx}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 92741 Answers: 0 Comments: 9

lim_(x→0) ((2/(sin^2 (x))) + (1/(ln(cos (x))))) =

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{2}}{\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:+\:\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{cos}\:\left(\mathrm{x}\right)\right)}\right)\:= \\ $$

Question Number 92743 Answers: 0 Comments: 2

y − y_1 = m ( x − x_1 ) ⇒ y − 1 = 5 ( x − 2 ) y − 1 = 5x − 10 y = 5x − 10 + 1 ∴ y = 5x − 9

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:\:−\:\:{y}_{\mathrm{1}} \:\:=\:\:{m}\:\left(\:{x}\:−\:{x}_{\mathrm{1}} \:\right) \\ $$$$\:\:\:\Rightarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:−\:\mathrm{1}\:=\:\:\mathrm{5}\:\left(\:{x}\:−\:\mathrm{2}\:\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:−\:\mathrm{1}\:=\:\:\mathrm{5}{x}\:−\:\mathrm{10} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:=\:\:\mathrm{5}{x}\:−\:\mathrm{10}\:+\:\mathrm{1} \\ $$$$\:\:\:\:\therefore\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:=\:\:\mathrm{5}{x}\:−\:\mathrm{9} \\ $$$$\: \\ $$

Question Number 92740 Answers: 0 Comments: 0

∫ ((ln(x))/(cos x)) dx ?

$$\int\:\frac{\mathrm{ln}\left({x}\right)}{\mathrm{cos}\:{x}}\:{dx}\:? \\ $$

Question Number 92729 Answers: 1 Comments: 0

if p_n is the product of the terms in the nth row of the pascal′s triangle find lim_(n→∞) ((p_(n−1) p_(n+1) )/((p_n )^2 ))

$${if}\:{p}_{{n}} \:{is}\:{the}\:{product}\:{of}\:{the}\:{terms}\:{in} \\ $$$${the}\:{nth}\:{row}\:{of}\:{the}\:{pascal}'{s}\:{triangle} \\ $$$${find} \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\frac{{p}_{{n}−\mathrm{1}} {p}_{{n}+\mathrm{1}} }{\left({p}_{{n}} \right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 92727 Answers: 1 Comments: 0

Solve: x^y = y^x ....... (i) 3^x = 15^y ...... (ii) x ≠ y, x, y ∈ R

$$\mathrm{Solve}:\:\:\:\:\:\mathrm{x}^{\mathrm{y}} \:\:=\:\:\mathrm{y}^{\mathrm{x}} \:\:\:\:\:.......\:\:\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}^{\mathrm{x}} \:\:=\:\:\mathrm{15}^{\mathrm{y}} \:\:\:\:......\:\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\mathrm{x}\:\:\neq\:\:\mathrm{y},\:\:\:\:\:\:\:\mathrm{x},\:\:\mathrm{y}\:\in\:\mathbb{R} \\ $$

Question Number 92723 Answers: 1 Comments: 0

Question Number 92717 Answers: 2 Comments: 5

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