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let f(x)=2(√(3−x)) and g(x) =x^2 −2x +5 1) calculate fog(x) and determine D_(fog) 2) calculate ∫ fog(x)dx 3) calculate ∫ ((f^(−1) (x))/(f(x)))dx and ∫ ((f^(−1) og(x))/(fog(x)))dx

$${let}\:{f}\left({x}\right)=\mathrm{2}\sqrt{\mathrm{3}−{x}}\:{and}\:{g}\left({x}\right)\:={x}^{\mathrm{2}} −\mathrm{2}{x}\:+\mathrm{5} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{fog}\left({x}\right)\:{and}\:{determine}\:{D}_{{fog}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int\:{fog}\left({x}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int\:\frac{{f}^{−\mathrm{1}} \left({x}\right)}{{f}\left({x}\right)}{dx}\:\:{and}\:\:\int\:\frac{{f}^{−\mathrm{1}} {og}\left({x}\right)}{{fog}\left({x}\right)}{dx} \\ $$

Question Number 93892 Answers: 0 Comments: 3

let a_(n+1) =(√(2+(√a_n ))) a_0 =(√2) find lim_(n→∞) a_(n+1)

$${let} \\ $$$${a}_{{n}+\mathrm{1}} =\sqrt{\mathrm{2}+\sqrt{{a}_{{n}} }}\:\:\:\:\:\:{a}_{\mathrm{0}} =\sqrt{\mathrm{2}} \\ $$$${find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}+\mathrm{1}} \\ $$

Question Number 93874 Answers: 1 Comments: 4

find k if the vector (1^ −2,k) in R^3 be a linear combination of the vectors (3,0,2) &(2,−1,−5)

$$\mathrm{find}\:\mathrm{k}\:\mathrm{if}\:\mathrm{the}\:\mathrm{vector}\:\left(\bar {\mathrm{1}}−\mathrm{2},\mathrm{k}\right)\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\mathrm{be}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{combination}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{vectors}\:\left(\mathrm{3},\mathrm{0},\mathrm{2}\right)\:\&\left(\mathrm{2},−\mathrm{1},−\mathrm{5}\right)\: \\ $$

Question Number 93873 Answers: 1 Comments: 1

find ∫_(−∞) ^∞ (dx/((x^2 +2x+4)^3 ))

$${find}\:\int_{−\infty} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}\right)^{\mathrm{3}} } \\ $$

Question Number 93869 Answers: 0 Comments: 2

∫ (dx/(1−tan^2 x)) ?

$$\int\:\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:? \\ $$

Question Number 93860 Answers: 0 Comments: 5

Consider the progression (I_n )_(n∈N) where I_n =∫_0 ^1 ((sin(πt))/(t+n))dt 1\ Show that: ∀n≥0, 0≤I_(n+1) ≤I_n and deduce that the series is convergent. 2\ Show that 0≤I_n ≤ln(((n+1)/n)) and deduce the limit the series (I_n )_(n∈N) 3\ Calculate lim_(n→+∞) (nI_n )

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{progression}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{where}\:\mathrm{I}_{\mathrm{n}} \:=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{sin}\left(\pi\mathrm{t}\right)}{\mathrm{t}+\mathrm{n}}\mathrm{dt} \\ $$$$\mathrm{1}\backslash\:\mathrm{Show}\:\mathrm{that}:\:\forall\mathrm{n}\geqslant\mathrm{0},\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}+\mathrm{1}} \leqslant\mathrm{I}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{series}\:\mathrm{is}\:\mathrm{convergent}. \\ $$$$\mathrm{2}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\mathrm{ln}\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{n}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{limit} \\ $$$$\mathrm{the}\:\mathrm{series}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \\ $$$$\mathrm{3}\backslash\:\mathrm{Calculate}\:\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\left(\mathrm{nI}_{\mathrm{n}} \right) \\ $$

Question Number 93859 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (1/(n(n+1)))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)} \\ $$

Question Number 93855 Answers: 1 Comments: 0

At what point of the parabola y=x^2 will the tangent a\ be parallel to the line y=4x−5 b\ be perpendicular to 2x−6y+5=0 c\ make an angle of 45^° with 3x−y+1=0

$$\mathrm{At}\:\mathrm{what}\:\mathrm{point}\:\mathrm{of}\:\mathrm{the}\:\mathrm{parabola}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} \\ $$$$\mathrm{will}\:\mathrm{the}\:\mathrm{tangent}\: \\ $$$$\mathrm{a}\backslash\:\mathrm{be}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=\mathrm{4x}−\mathrm{5} \\ $$$$\mathrm{b}\backslash\:\mathrm{be}\:\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{2x}−\mathrm{6y}+\mathrm{5}=\mathrm{0} \\ $$$$\mathrm{c}\backslash\:\mathrm{make}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{45}^{°} \:\mathrm{with}\:\mathrm{3x}−\mathrm{y}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 93853 Answers: 1 Comments: 0

Find the angle at which the parabola y=x^2 cuts through the line 3x−y−2=0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{at}\:\mathrm{which}\:\mathrm{the}\:\mathrm{parabola} \\ $$$$\mathrm{y}=\mathrm{x}^{\mathrm{2}} \:\mathrm{cuts}\:\mathrm{through}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}−\mathrm{y}−\mathrm{2}=\mathrm{0} \\ $$

Question Number 93847 Answers: 0 Comments: 2

log _5 (x−2)+log _8 (x−4)= log _6 (x−1)

$$\mathrm{log}\:_{\mathrm{5}} \:\left(\mathrm{x}−\mathrm{2}\right)+\mathrm{log}\:_{\mathrm{8}} \:\left(\mathrm{x}−\mathrm{4}\right)=\:\mathrm{log}\:_{\mathrm{6}} \left(\mathrm{x}−\mathrm{1}\right) \\ $$

Question Number 93844 Answers: 1 Comments: 0

let f(x)=(√x) and g(x)=(√x) find the domain of (f.g)(x) ? help me sir

$${let}\:{f}\left({x}\right)=\sqrt{{x}}\:\:{and}\:{g}\left({x}\right)=\sqrt{{x}}\:{find}\:{the}\:{domain}\:{of}\:\left({f}.{g}\right)\left({x}\right)\:? \\ $$$${help}\:{me}\:{sir} \\ $$

Question Number 93843 Answers: 1 Comments: 0

Use Horner to solve it . (a^2 −5ab−3b^2 ) : (a−b) find the remainder .

$${Use}\:\:{Horner}\:\:{to}\:\:{solve}\:{it}\:. \\ $$$$\:\:\left({a}^{\mathrm{2}} −\mathrm{5}{ab}−\mathrm{3}{b}^{\mathrm{2}} \right)\::\:\left({a}−{b}\right) \\ $$$${find}\:\:{the}\:\:{remainder}\:\:. \\ $$

Question Number 93837 Answers: 0 Comments: 5

Question Number 93836 Answers: 0 Comments: 0

If cosα+cosβ+cosγ=0 = sinα+sinβ+sinγ, prove that (i) cos3α+cos3β+cosγ = 3cos(a+β+γ) (ii) sin3α+sin3β+sinγ = 3sin(α+β+γ) (iii) cos2α+cos2β+cos2γ = 0 (iv) sin2α+sin2β+sin2γ = 0 (Hints: Take a=cis α, b=cis β, c=cis γ, a+b+c=0 ⇒ a^3 +b^3 +c^3 =3abc (1/a)+(1/b)+(1/c)=0 ⇒ a^2 +b^2 +c^2 =0) (v) cos^2 α+cos^2 β+cosγ = sin^2 α+sin^2 β+sin^2 γ = (3/2).

$${If}\:{cos}\alpha+{cos}\beta+{cos}\gamma=\mathrm{0}\:=\:{sin}\alpha+{sin}\beta+{sin}\gamma,\:{prove}\:{that} \\ $$$$\left(\mathrm{i}\right)\:{cos}\mathrm{3}\alpha+{cos}\mathrm{3}\beta+{cos}\gamma\:=\:\mathrm{3}{cos}\left({a}+\beta+\gamma\right) \\ $$$$\left(\mathrm{ii}\right)\:{sin}\mathrm{3}\alpha+{sin}\mathrm{3}\beta+{sin}\gamma\:=\:\mathrm{3}{sin}\left(\alpha+\beta+\gamma\right) \\ $$$$\left(\mathrm{iii}\right)\:{cos}\mathrm{2}\alpha+{cos}\mathrm{2}\beta+{cos}\mathrm{2}\gamma\:=\:\mathrm{0} \\ $$$$\left(\mathrm{iv}\right)\:{sin}\mathrm{2}\alpha+{sin}\mathrm{2}\beta+{sin}\mathrm{2}\gamma\:=\:\mathrm{0} \\ $$$$\left(\mathrm{Hints}:\:\mathrm{T}{ake}\:{a}=\mathrm{cis}\:\alpha,\:\mathrm{b}=\mathrm{cis}\:\beta,\:\mathrm{c}=\mathrm{cis}\:\gamma,\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{0}\:\Rightarrow\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} =\mathrm{3abc}\right. \\ $$$$\left.\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}=\mathrm{0}\:\Rightarrow\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} =\mathrm{0}\right) \\ $$$$\left(\mathrm{v}\right)\:{cos}^{\mathrm{2}} \alpha+{cos}^{\mathrm{2}} \beta+{cos}\gamma\:=\:{sin}^{\mathrm{2}} \alpha+{sin}^{\mathrm{2}} \beta+{sin}^{\mathrm{2}} \gamma\:=\:\frac{\mathrm{3}}{\mathrm{2}}. \\ $$

Question Number 93828 Answers: 1 Comments: 2

sin x ((dy/dx))−y = 2sin x

$$\mathrm{sin}\:\mathrm{x}\:\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)−\mathrm{y}\:=\:\mathrm{2sin}\:\mathrm{x}\: \\ $$

Question Number 93827 Answers: 0 Comments: 1

simplify: (((cosα+isinα)^3 )/((sinβ+icosβ)^4 ))

$${simplify}:\:\frac{\left({cos}\alpha+\boldsymbol{\mathrm{i}}{sin}\alpha\right)^{\mathrm{3}} }{\left({sin}\beta+\boldsymbol{\mathrm{i}}{cos}\beta\right)^{\mathrm{4}} } \\ $$

Question Number 93824 Answers: 1 Comments: 0

Question Number 93820 Answers: 1 Comments: 1

∫ ((x^2 dx)/(√(x^2 −x+1)))

$$\int\:\frac{{x}^{\mathrm{2}} \:{dx}}{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\: \\ $$

Question Number 93818 Answers: 1 Comments: 0

(y−xy^2 )dx +(x+x^2 y^2 )dy = 0

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

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