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Question Number 60503 Answers: 1 Comments: 0

calculate Σ_(n=1) ^∞ ((1+2+3+...+n)/(1^3 +2^3 +3^3 +...+n^3 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+{n}}{\mathrm{1}^{\mathrm{3}} \:+\mathrm{2}^{\mathrm{3}} \:+\mathrm{3}^{\mathrm{3}} \:+...+{n}^{\mathrm{3}} } \\ $$

Question Number 60502 Answers: 0 Comments: 2

let f(x) =arctan(2x) ln (1−x^2 ) 1) calculate f^′ (x) 2) determine f^((n)) (x) and f^((n)) (0) 3) developp f at integr serie .

$${let}\:{f}\left({x}\right)\:={arctan}\left(\mathrm{2}{x}\right)\:{ln}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 60501 Answers: 1 Comments: 1

let A = ((( 1 1)),((1 1)) ) 1)calculate A^n 2) determine e^A and e^(−A) .

$${let}\:{A}\:=\begin{pmatrix}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{e}^{{A}} \:\:\:{and}\:{e}^{−{A}} \:. \\ $$$$ \\ $$

Question Number 60500 Answers: 0 Comments: 2

let A = (((1 1)),((−2 3)) ) 1) find A^(−1) 2) calculate A^n 3) determine e^A and e^(−2A) .

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{−\mathrm{2}\:\:\:\mathrm{3}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{e}^{{A}} \:\:\:{and}\:{e}^{−\mathrm{2}{A}} \:. \\ $$

Question Number 60499 Answers: 0 Comments: 2

find the value of Σ_(n=1) ^∞ (((−1)^n )/(n^3 (n+1)^4 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 60498 Answers: 0 Comments: 4

let f(t) =∫_0 ^3 (√(t +x +x^2 ))dx with t ≥(1/4) 1) find a explicit form of f(t) 2) find also g(t) = ∫_0 ^3 (dx/(√(t+x +x^2 ))) 3) calculate ∫_0 ^3 (√(1+x+x^2 ))dx , ∫_0 ^3 (√(2 +x+x^2 ))dx ∫_0 ^3 (dx/(√(2+x +x^2 ))) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{{t}\:+{x}\:+{x}^{\mathrm{2}} }{dx}\:\:{with}\:{t}\:\geqslant\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \:\:\:\frac{{dx}}{\sqrt{{t}+{x}\:+{x}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{3}} \:\sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }{dx}\:,\:\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{2}\:+{x}+{x}^{\mathrm{2}} }{dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{3}} \:\:\:\frac{{dx}}{\sqrt{\mathrm{2}+{x}\:+{x}^{\mathrm{2}} }}\:\:. \\ $$

Question Number 60595 Answers: 0 Comments: 2

let f(a) =∫_0 ^1 ((ln^2 (x))/((1−ax)^2 )) dx with ∣a∣<1 1) find a explicit form of f(a) 2) determine A(θ) =∫_0 ^1 ((ln^2 (x))/((1−(cosθ)x)^2 ))dx with 0<θ<(π/2)

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−{ax}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−\left({cos}\theta\right){x}\right)^{\mathrm{2}} }{dx}\:\:{with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 60496 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((lnx)/((1−x)^2 ))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{lnx}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 60495 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(x))/((1−x^2 )^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 60494 Answers: 1 Comments: 1

find ∫ (√((√(2+x^2 ))−x))dx

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

Question Number 60493 Answers: 0 Comments: 0

let f(x)=(√(1+(√(1+x^2 )))) approximate f(x) by a polynome at v(0)

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$$${approximate}\:{f}\left({x}\right)\:{by}\:{a}\:{polynome} \\ $$$${at}\:{v}\left(\mathrm{0}\right) \\ $$

Question Number 60484 Answers: 1 Comments: 2

If a + b + c = 4 then find a^3 + b^3 + c^(3 ) = ?

$${If}\:\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{4} \\ $$$$ \\ $$$${then}\:{find} \\ $$$${a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:+\:{c}^{\mathrm{3}\:} =\:? \\ $$

Question Number 60481 Answers: 0 Comments: 0

Question Number 60856 Answers: 0 Comments: 0

if 0<x<1,lim_(n→+∞) ((x^x^x^.^.^.^x }n)/(((x^x )^x )^(x...}n) ))=? (? can be expressed by x)

$${if}\:\:\mathrm{0}<{x}<\mathrm{1},\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\frac{\left.{x}^{{x}^{{x}^{.^{.^{.^{{x}} } } } } } \right\}{n}}{\left(\left({x}^{{x}} \right)^{{x}} \right)^{\left.{x}...\right\}{n}} }=? \\ $$$$\left(?\:{can}\:{be}\:{expressed}\:{by}\:{x}\right) \\ $$

Question Number 60477 Answers: 2 Comments: 0

Question Number 60475 Answers: 1 Comments: 0

Question Number 60461 Answers: 1 Comments: 0

Two cogged wheels, of which one has 16 cogs and other has 27, work into each other. If the latter turns 80 times in three quarters of a minute, how often does the other turn in 8 seconds?

$$\mathrm{Two}\:\mathrm{cogged}\:\mathrm{wheels},\:\mathrm{of}\:\mathrm{which}\:\mathrm{one}\:\mathrm{has} \\ $$$$\mathrm{16}\:\mathrm{cogs}\:\mathrm{and}\:\mathrm{other}\:\mathrm{has}\:\mathrm{27},\:\mathrm{work}\:\mathrm{into}\: \\ $$$$\mathrm{each}\:\mathrm{other}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{latter}\:\mathrm{turns}\:\mathrm{80}\:\mathrm{times}\:\mathrm{in}\: \\ $$$$\mathrm{three}\:\mathrm{quarters}\:\mathrm{of}\:\mathrm{a}\:\mathrm{minute},\:\mathrm{how}\:\mathrm{often} \\ $$$$\mathrm{does}\:\mathrm{the}\:\mathrm{other}\:\mathrm{turn}\:\mathrm{in}\:\mathrm{8}\:\mathrm{seconds}? \\ $$

Question Number 60459 Answers: 0 Comments: 4

Question Number 60453 Answers: 0 Comments: 0

Question Number 60452 Answers: 1 Comments: 1

Question Number 60450 Answers: 0 Comments: 0

Question Number 60445 Answers: 2 Comments: 1

Question Number 60441 Answers: 1 Comments: 0

If a sum of money doubles itself in a time T, when compounded continuously, find the rate of interest, in terms of T.

$$\mathrm{If}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\:\mathrm{money}\:\mathrm{doubles}\:\mathrm{itself} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{time}\:\mathrm{T},\:\mathrm{when}\:\mathrm{compounded} \\ $$$$\mathrm{continuously},\:\mathrm{find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of} \\ $$$$\mathrm{interest},\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{T}. \\ $$

Question Number 60426 Answers: 0 Comments: 2

the function is considered f(x,y)=e^(xy) +(x/y)+sen((2x+3y)π) Calcule: (∂f/∂x),(∂f/∂y),(∂^2 f/∂x^2 ),(∂^2 f/(∂x∂y)). f_x (0,1),f_y (2,−1), f_(xx) (0,1),f_(xy) (2,−1)

$${the}\:{function}\:{is}\:{considered}\: \\ $$$${f}\left({x},{y}\right)={e}^{{xy}} +\frac{{x}}{{y}}+{sen}\left(\left(\mathrm{2}{x}+\mathrm{3}{y}\right)\pi\right)\:{Calcule}: \\ $$$$\frac{\partial{f}}{\partial{x}},\frac{\partial{f}}{\partial{y}},\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} },\frac{\partial^{\mathrm{2}} {f}}{\partial{x}\partial{y}}.\:\:\:{f}_{{x}} \left(\mathrm{0},\mathrm{1}\right),{f}_{{y}} \left(\mathrm{2},−\mathrm{1}\right),\:{f}_{{xx}} \left(\mathrm{0},\mathrm{1}\right),{f}_{{xy}} \left(\mathrm{2},−\mathrm{1}\right) \\ $$

Question Number 60425 Answers: 0 Comments: 0

Question Number 60424 Answers: 0 Comments: 2

let z ∈C and ∣z∣<1 find f(x)=∫_0 ^1 ln(1+zx)dx.

$${let}\:{z}\:\in{C}\:{and}\:\:\mid{z}\mid<\mathrm{1}\:\:{find} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{zx}\right){dx}. \\ $$

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