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Question Number 76431    Answers: 1   Comments: 1

how to find x^(2048) + x^(−2048) if x+x^(−1) =(((√5)+1)/2)

$${how}\:{to}\:{find}\:{x}^{\mathrm{2048}} \:+\:{x}^{−\mathrm{2048}} \\ $$$${if}\:\:{x}+{x}^{−\mathrm{1}} =\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 76428    Answers: 2   Comments: 0

Question Number 76424    Answers: 1   Comments: 1

how to calculate ∫(√(x/(√(x/(√(x/(√(...)))))))) dx ?

$${how}\:{to}\:{calculate}\:\int\sqrt{\frac{{x}}{\sqrt{\frac{{x}}{\sqrt{\frac{{x}}{\sqrt{...}}}}}}}\:\:{dx}\:? \\ $$

Question Number 76407    Answers: 0   Comments: 0

Question Number 76404    Answers: 0   Comments: 2

Hello have nice end of year good bless you all i respond note in y re message becsuse i have so many problemes that mack me feel no pleasur any more to do somthing i think its importante to say it i will back Soon i hop so Sorry for my English

$$\mathrm{Hello}\:\mathrm{have}\:\mathrm{nice}\:\mathrm{end}\:\mathrm{of}\:\mathrm{year}\:\mathrm{good}\:\mathrm{bless}\:\mathrm{you} \\ $$$$\mathrm{all}\:\mathrm{i}\:\mathrm{respond}\:\mathrm{note}\:\mathrm{in}\:\mathrm{y}\:\mathrm{re}\:\mathrm{message}\:\mathrm{becsuse}\:\mathrm{i}\:\mathrm{have}\:\mathrm{so}\:\mathrm{many}\:\mathrm{problemes} \\ $$$$\mathrm{that}\:\mathrm{mack}\:\mathrm{me}\:\mathrm{feel}\:\mathrm{no}\:\mathrm{pleasur}\:\mathrm{any}\:\mathrm{more}\:\mathrm{to}\:\mathrm{do}\:\mathrm{somthing} \\ $$$$\mathrm{i}\:\mathrm{think}\:\mathrm{its}\:\mathrm{importante}\:\mathrm{to}\:\mathrm{say}\:\mathrm{it}\:\mathrm{i}\:\mathrm{will}\:\mathrm{back}\:\mathrm{Soon}\:\mathrm{i}\:\mathrm{hop}\:\mathrm{so}\:\:\mathrm{Sorry}\:\mathrm{for} \\ $$$$\mathrm{my}\:\mathrm{English} \\ $$

Question Number 76401    Answers: 1   Comments: 0

In a quadratic equation ax^2 −bx+c=0, a,b, c are distinct primes and the product of the sum of the roots and product of the roots is ((91)/9) . Find the absolute value of difference between the sum of the roots and the product of the roots.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{equation}\:{ax}^{\mathrm{2}} −{bx}+{c}=\mathrm{0}, \\ $$$${a},{b},\:{c}\:\:\mathrm{are}\:\mathrm{distinct}\:\mathrm{primes}\:\mathrm{and}\:\mathrm{the}\:\mathrm{product} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{and}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{is}\:\frac{\mathrm{91}}{\mathrm{9}}\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{absolute}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{difference}\:\mathrm{between}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}. \\ $$

Question Number 76397    Answers: 0   Comments: 2

solve for z∈C [z=a+bi; z^ =a−bi; r∈R] (√(r^2 −z^2 ))=z^ (√(r^2 +z^2 ))=z^

$$\mathrm{solve}\:\mathrm{for}\:{z}\in\mathbb{C} \\ $$$$\left[{z}={a}+{b}\mathrm{i};\:\bar {{z}}={a}−{b}\mathrm{i};\:{r}\in\mathbb{R}\right] \\ $$$$\sqrt{{r}^{\mathrm{2}} −{z}^{\mathrm{2}} }=\bar {{z}} \\ $$$$\sqrt{{r}^{\mathrm{2}} +{z}^{\mathrm{2}} }=\bar {{z}} \\ $$

Question Number 76386    Answers: 1   Comments: 2

Are give two parallel lines and a point A is given between them, and the distance from A to the lines are a and b. Determine the cathetus of right−angled triangle in A knowing that the other vertices belong to the parallel lines and the area of the triangle is equal to k^2 .

$${Are}\:{give}\:{two}\:{parallel}\:{lines}\:{and}\:{a}\:{point} \\ $$$${A}\:{is}\:{given}\:{between}\:{them},\:{and}\:{the} \\ $$$${distance}\:{from}\:{A}\:{to}\:{the}\:{lines}\:{are}\: \\ $$$$\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}.\:{Determine}\:{the}\:{cathetus}\:{of} \\ $$$${right}−{angled}\:{triangle}\:{in}\:{A}\:{knowing} \\ $$$${that}\:{the}\:{other}\:{vertices}\:{belong}\:{to} \\ $$$${the}\:{parallel}\:{lines}\:{and}\:{the}\:{area}\:{of} \\ $$$${the}\:{triangle}\:{is}\:{equal}\:{to}\:\boldsymbol{{k}}^{\mathrm{2}} . \\ $$

Question Number 76384    Answers: 0   Comments: 1

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Question Number 76368    Answers: 3   Comments: 5

prove that 1. Σ_(r=1) ^n r = (1/2)n(n+1) 2. Σ_(r=1) ^n r^2 = (1/6)n(n+1)(2n + 1) 3. Σ_(r=1) ^n r^3 = (1/4)n^2 (n + 1)^2

$${prove}\:{that} \\ $$$$\mathrm{1}.\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right) \\ $$$$\mathrm{2}.\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{r}^{\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{6}}{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}\:+\:\mathrm{1}\right) \\ $$$$\mathrm{3}.\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{3}} =\:\frac{\mathrm{1}}{\mathrm{4}}{n}^{\mathrm{2}} \left({n}\:+\:\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 76367    Answers: 0   Comments: 4

prove that Σ_(r=1) ^∞ (1/r^2 ) = (π^2 /6)

$${prove}\:{that}\:\:\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$

Question Number 76361    Answers: 1   Comments: 3

let A = (((1 1)),((1 1)) ) 1) calculate A^n 2) find e^A ,e^(−A) 3) find sinA and cosA 4) find ch(A) and sh(A)

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{e}^{{A}} \:\:,{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{sinA}\:{and}\:{cosA} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{ch}\left({A}\right)\:{and}\:{sh}\left({A}\right) \\ $$

Question Number 76360    Answers: 0   Comments: 1

calculate ∫_1 ^∞ (dx/(x^3 (x^2 −2x+3)))

$${calculate}\:\:\int_{\mathrm{1}} ^{\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{3}\right)} \\ $$

Question Number 76359    Answers: 0   Comments: 1

calculate ∫_(−∞) ^∞ (dx/((x^2 −3x+7)^2 ))

$${calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{7}\right)^{\mathrm{2}} } \\ $$

Question Number 76356    Answers: 0   Comments: 1

find f(x) =∫ (dt/(√(t^2 −xt +1))) with x real.

$${find}\:{f}\left({x}\right)\:=\int\:\:\frac{{dt}}{\sqrt{{t}^{\mathrm{2}} −{xt}\:+\mathrm{1}}}\:\:{with}\:{x}\:{real}. \\ $$

Question Number 76355    Answers: 2   Comments: 0

find ∫ ((arctan((√(1+x))))/(2+x))dx

$${find}\:\int\:\frac{{arctan}\left(\sqrt{\mathrm{1}+{x}}\right)}{\mathrm{2}+{x}}{dx} \\ $$

Question Number 76354    Answers: 0   Comments: 0

calculate Σ_(n=1) ^∞ (−1)^n arctan((1/(n^2 +n)))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:{arctan}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+{n}}\right) \\ $$

Question Number 76353    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ ((arctan(sin(πx^2 )))/(x^2 +π^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({sin}\left(\pi{x}^{\mathrm{2}} \right)\right)}{{x}^{\mathrm{2}} \:+\pi^{\mathrm{2}} }{dx} \\ $$

Question Number 76352    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((sin(arctan(x^2 +2)))/(x^2 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({arctan}\left({x}^{\mathrm{2}} +\mathrm{2}\right)\right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 76351    Answers: 0   Comments: 0

calculate U_n =∫_(1/n) ^1 Γ(x)dx and find lim_(n→+∞) U_n

$${calculate}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\Gamma\left({x}\right){dx}\:\:{and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 76350    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((arctan(e^x^2 ))/(x^2 +4))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({e}^{{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 76346    Answers: 1   Comments: 5

Question Number 76341    Answers: 0   Comments: 1

Question Number 76340    Answers: 3   Comments: 1

Question Number 76328    Answers: 0   Comments: 2

Question Number 76326    Answers: 2   Comments: 0

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