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Question Number 55051 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^1 (dx/(1+x+x^2 +...+x^n ))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+...+{x}^{{n}} } \\ $$

Question Number 55454 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ ((ln(x))/(x^2 +a)) with a>0 1) calculate f(a) intermsof a 2) find the values of ∫_0 ^∞ ((ln(x))/(x^2 +1))dx and ∫_0 ^∞ ((ln(x))/(x^2 +2))dx 3) let g(a) =∫_0 ^∞ ((ln(x))/((x^2 +a)^n )) dx .calculate g(a) interms of a 4) find values> of ∫_0 ^∞ ((ln(x))/((x^2 +3)^2 ))dx 5) find nature of the serie Σ f(n) andΣ g(n)

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+{a}}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intermsof}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{2}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({x}\right)}{\left({x}^{\mathrm{2}} \:+{a}\right)^{{n}} }\:{dx}\:\:.{calculate}\:{g}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{values}>\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{f}\left({n}\right)\:\:{and}\Sigma\:{g}\left({n}\right)\: \\ $$

Question Number 55044 Answers: 1 Comments: 2

Question Number 55039 Answers: 1 Comments: 3

α and β,are 2 roots of eq: ax^2 +bx+c=0 with conditions: { ((α^2 =β+b)),((β^2 =α+a)) :} find: c in terms of: a and b.

$$\alpha\:{and}\:\beta,{are}\:\mathrm{2}\:{roots}\:{of}\:{eq}: \\ $$$$\:\:\:\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0}\:{with}\:{conditions}: \\ $$$$\:\:\:\:\begin{cases}{\alpha^{\mathrm{2}} =\beta+{b}}\\{\beta^{\mathrm{2}} =\alpha+{a}}\end{cases} \\ $$$${find}:\:\:\boldsymbol{{c}}\:{in}\:{terms}\:{of}:\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}. \\ $$

Question Number 55030 Answers: 1 Comments: 5

Find lim_(n→∞) (∫_0 ^1 (ln x)^n dx)

$$\mathrm{Find} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{ln}\:{x}\right)^{{n}} \:{dx}\right) \\ $$

Question Number 55006 Answers: 1 Comments: 0

Question Number 55011 Answers: 0 Comments: 1

Σ_(k=o) ^(n−1) (1/(2−x^k )) find out the summetion

$$\underset{{k}={o}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{\mathrm{2}−{x}^{{k}} }\:\:\:\:{find}\:{out}\:{the}\:{summetion} \\ $$$$ \\ $$

Question Number 54995 Answers: 1 Comments: 3

∫ ((x^3 +x^2 ))^(1/3) dx

$$\int\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 54992 Answers: 2 Comments: 0

lim_(n→∞) ((√(n^3 +n^2 +n+1))−n)

$$\underset{{n}\rightarrow\infty} {{lim}}\left(\sqrt{{n}^{\mathrm{3}} +{n}^{\mathrm{2}} +{n}+\mathrm{1}}−{n}\right) \\ $$

Question Number 54991 Answers: 0 Comments: 3

Find all the roots of: z^4 + 16i = 0

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}:\:\:\:\:\:\:\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{16i}\:\:=\:\:\mathrm{0} \\ $$

Question Number 55014 Answers: 3 Comments: 0

what is the value of t that makes x^2 +10x+t a perfect square?

$${what}\:{is}\:{the}\:{value}\:{of}\:{t}\:{that}\:{makes}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{10}{x}+{t}\:{a}\:{perfect}\:{square}? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 54983 Answers: 1 Comments: 2

Find shortest distance from origin to the cubic y=x^3 −10x^2 +27x−18.

$${Find}\:{shortest}\:{distance}\:{from}\:{origin} \\ $$$${to}\:{the}\:{cubic}\:\:\:{y}={x}^{\mathrm{3}} −\mathrm{10}{x}^{\mathrm{2}} +\mathrm{27}{x}−\mathrm{18}. \\ $$

Question Number 54975 Answers: 3 Comments: 0

f(x) = x^3 − 2x + 4 f^(−1) (x) = ?

$${f}\left({x}\right)\:\:=\:\:{x}^{\mathrm{3}} \:−\:\mathrm{2}{x}\:+\:\mathrm{4} \\ $$$${f}\:^{−\mathrm{1}} \left({x}\right)\:\:=\:\:? \\ $$$$ \\ $$

Question Number 54972 Answers: 1 Comments: 0

∫_0 ^π ∫_0 ^(4cos z) ∫_0 ^(√(16−y^2 )) ydxdydz

$$\int_{\mathrm{0}} ^{\pi} \int_{\mathrm{0}} ^{\mathrm{4cos}\:{z}} \int_{\mathrm{0}} ^{\sqrt{\mathrm{16}−{y}^{\mathrm{2}} }} {ydxdydz} \\ $$

Question Number 54971 Answers: 2 Comments: 0

∫_0 ^(π/2) (((√(tanx)) sin^2 x)/((√(sin xcos x)) tan x))dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\sqrt{{tanx}}\:\mathrm{sin}\:^{\mathrm{2}} {x}}{\sqrt{\mathrm{sin}\:{x}\mathrm{cos}\:{x}}\:\mathrm{tan}\:{x}}{dx} \\ $$

Question Number 54970 Answers: 1 Comments: 0

If a cos A=b cos B in a △ABC, then ∠C is

$$\mathrm{If}\:\:\:{a}\:\mathrm{cos}\:{A}={b}\:\mathrm{cos}\:{B}\:\:\mathrm{in}\:\mathrm{a}\:\bigtriangleup{ABC},\: \\ $$$$\mathrm{then}\:\angle{C}\:\:\mathrm{is} \\ $$

Question Number 54969 Answers: 0 Comments: 0

Let λ is value of characteristic matrices P the fill P^t =P^2 find all λ the posible

$$\mathrm{Let}\:\lambda\:\mathrm{is}\:\mathrm{value}\:\mathrm{of}\:\mathrm{characteristic} \\ $$$$\mathrm{matrices}\:{P}\:\:\mathrm{the}\:\mathrm{fill}\:{P}^{{t}} ={P}^{\mathrm{2}} \\ $$$$\mathrm{find}\:\mathrm{all}\:\lambda\:\mathrm{the}\:\mathrm{posible} \\ $$

Question Number 54968 Answers: 1 Comments: 0

Let A matrices order 2×2 the fill tr(A^2 )=[tr(A)]^2 a. Find det(A) b. If A can′t diagonalizing, find tr(A)

$$\mathrm{Let}\:{A}\:\mathrm{matrices}\:\mathrm{order}\:\mathrm{2}×\mathrm{2}\:\mathrm{the}\:\mathrm{fill} \\ $$$$\mathrm{tr}\left({A}^{\mathrm{2}} \right)=\left[{tr}\left({A}\right)\right]^{\mathrm{2}} \\ $$$$\mathrm{a}.\:\mathrm{Find}\:\mathrm{det}\left(\mathrm{A}\right) \\ $$$$\mathrm{b}.\:\mathrm{If}\:\mathrm{A}\:\mathrm{can}'\mathrm{t}\:\mathrm{diagonalizing},\:\mathrm{find}\:\mathrm{tr}\left({A}\right) \\ $$$$ \\ $$

Question Number 54967 Answers: 0 Comments: 2

Let x_1 , x_2 , x_3 the number real x_1 <x_2 <x_3 . T : P_2 →R^3 defined with rule T= [((P(x_1 ))),((P(x_2 ))),((P(x_3 ))) ] for all P(x) ∈ P_2 a) Prove that T form linear transformation b) check whether T bijektive

$$\mathrm{Let}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} \:\mathrm{the}\:\mathrm{number}\:\mathrm{real} \\ $$$${x}_{\mathrm{1}} <{x}_{\mathrm{2}} <{x}_{\mathrm{3}} .\:{T}\::\:{P}_{\mathrm{2}} \rightarrow{R}^{\mathrm{3}} \:\mathrm{defined} \\ $$$$\mathrm{with}\:\mathrm{rule}\:{T}=\begin{bmatrix}{{P}\left({x}_{\mathrm{1}} \right)}\\{{P}\left({x}_{\mathrm{2}} \right)}\\{{P}\left({x}_{\mathrm{3}} \right)}\end{bmatrix} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{P}\left({x}\right)\:\in\:{P}_{\mathrm{2}} \\ $$$$\left.{a}\right)\:{P}\mathrm{rove}\:\mathrm{that}\:{T}\:\:\mathrm{form}\:\mathrm{linear}\:\mathrm{transformation} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{check}\:\mathrm{whether}\:\:{T}\:\mathrm{bijektive} \\ $$

Question Number 54966 Answers: 1 Comments: 3

By differentiating x (√(1 + x)) with respect to x , Evaluate, ∫_( 0) ^( 2) (x/(√(1 + x))) dx

$$\mathrm{By}\:\mathrm{differentiating}\:\:\:\mathrm{x}\:\sqrt{\mathrm{1}\:+\:\mathrm{x}}\:\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{x}\:, \\ $$$$\mathrm{Evaluate},\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{2}} \:\frac{\mathrm{x}}{\sqrt{\mathrm{1}\:+\:\mathrm{x}}}\:\:\mathrm{dx} \\ $$

Question Number 55071 Answers: 1 Comments: 0

Question Number 55072 Answers: 0 Comments: 1

Question Number 54963 Answers: 0 Comments: 0

Let a number of real so matrices [(1,a,a),(a,1,a),(a,a,1) ]having three value characteristic real λ_1 ≥λ_2 ≥λ_3 >0 then a have to lies in interval...

$$\mathrm{Let}\:{a}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{so}\:\mathrm{matrices} \\ $$$$\begin{bmatrix}{\mathrm{1}}&{{a}}&{{a}}\\{{a}}&{\mathrm{1}}&{{a}}\\{{a}}&{{a}}&{\mathrm{1}}\end{bmatrix}\mathrm{having}\:\mathrm{three}\:\mathrm{value} \\ $$$$\mathrm{characteristic}\:\mathrm{real}\:\lambda_{\mathrm{1}} \geqslant\lambda_{\mathrm{2}} \geqslant\lambda_{\mathrm{3}} >\mathrm{0} \\ $$$$\mathrm{then}\:{a}\:\mathrm{have}\:\mathrm{to}\:\mathrm{lies}\:\mathrm{in}\:\mathrm{interval}... \\ $$

Question Number 54962 Answers: 0 Comments: 0

Let X={(−1, 0, 0), (1, 1, 0), (0, 1, 1) and ortogonal projection at X. Matrices representation P to basic basis in space Euclid R^3 is..

$$\mathrm{Let}\:{X}=\left\{\left(−\mathrm{1},\:\mathrm{0},\:\mathrm{0}\right),\:\left(\mathrm{1},\:\mathrm{1},\:\mathrm{0}\right),\:\left(\mathrm{0},\:\mathrm{1},\:\mathrm{1}\right)\right. \\ $$$$\mathrm{and}\:\mathrm{ortogonal}\:\mathrm{projection}\:\mathrm{at}\:{X}. \\ $$$$\mathrm{Matrices}\:\mathrm{representation}\:\mathrm{P}\:\mathrm{to} \\ $$$$\mathrm{basic}\:\mathrm{basis}\:\mathrm{in}\:\mathrm{space}\:\mathrm{Euclid}\:\mathrm{R}^{\mathrm{3}} \:\mathrm{is}.. \\ $$

Question Number 54955 Answers: 1 Comments: 0

Let vector set {u_1 , u_2 , u_3 , u_4 } in C^n free linear. So that {u_1 +αu_2 , u_2 +αu_3 , u_3 +αu_4 , u_4 +αu_1 } too free linear , scalar α have to...

$$\mathrm{Let}\:\mathrm{vector}\:\mathrm{set}\:\left\{{u}_{\mathrm{1}} ,\:{u}_{\mathrm{2}} ,\:{u}_{\mathrm{3}} ,\:{u}_{\mathrm{4}} \right\}\:\mathrm{in}\:\mathbb{C}^{{n}} \\ $$$$\mathrm{free}\:\mathrm{linear}.\:\mathrm{So}\:\mathrm{that}\: \\ $$$$\left\{{u}_{\mathrm{1}} +\alpha{u}_{\mathrm{2}} ,\:{u}_{\mathrm{2}} +\alpha{u}_{\mathrm{3}} ,\:{u}_{\mathrm{3}} +\alpha{u}_{\mathrm{4}} ,\:{u}_{\mathrm{4}} +\alpha{u}_{\mathrm{1}} \right\} \\ $$$$\mathrm{too}\:\mathrm{free}\:\mathrm{linear}\:,\:\mathrm{scalar}\:\alpha\:\mathrm{have}\:\mathrm{to}... \\ $$

Question Number 54953 Answers: 1 Comments: 1

The characteristic polynomial matrices [(1,(−2),3),(4,5,(−6)),((−7),8,9) ]is...

$$\mathrm{The}\:\mathrm{characteristic}\:\mathrm{polynomial} \\ $$$$\mathrm{matrices}\:\begin{bmatrix}{\mathrm{1}}&{−\mathrm{2}}&{\mathrm{3}}\\{\mathrm{4}}&{\mathrm{5}}&{−\mathrm{6}}\\{−\mathrm{7}}&{\mathrm{8}}&{\mathrm{9}}\end{bmatrix}\mathrm{is}... \\ $$

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