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Question Number 162622 Answers: 3 Comments: 0

Solve for real numbers: x^(12) - 15x^3 + 14 = 0

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{x}^{\mathrm{12}} \:-\:\mathrm{15x}^{\mathrm{3}} \:+\:\mathrm{14}\:=\:\mathrm{0} \\ $$

Question Number 162618 Answers: 2 Comments: 0

Calculate lim_(x→∞) (5^x +8^x )^(1/(3x))

$${Calculate} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{5}^{{x}} +\mathrm{8}^{{x}} \right)^{\frac{\mathrm{1}}{\mathrm{3}{x}}} \\ $$

Question Number 162604 Answers: 1 Comments: 0

I=∫_0 ^(π/4) xtg(x)dx=?

$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \boldsymbol{{xtg}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}=? \\ $$

Question Number 162598 Answers: 1 Comments: 0

Question Number 162589 Answers: 1 Comments: 0

Question Number 162585 Answers: 1 Comments: 0

lim_(n→∞) (2n∫_0 ^1 (x^n /(1+x^2 ))dx)^n =?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\right)^{\mathrm{n}} =? \\ $$

Question Number 162584 Answers: 1 Comments: 0

prove that ppcm(a,b)×pgcd(a,b)=∣ab∣

$${prove}\:{that} \\ $$$${ppcm}\left({a},{b}\right)×{pgcd}\left({a},{b}\right)=\mid{ab}\mid \\ $$

Question Number 162575 Answers: 2 Comments: 0

∫_0 ^π (x^2 /(1+sinx))dx

$$\int_{\mathrm{0}} ^{\pi} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{sin}{x}}{dx} \\ $$

Question Number 162561 Answers: 2 Comments: 0

Question Number 162560 Answers: 0 Comments: 1

Question Number 162552 Answers: 1 Comments: 1

𝛂_1 <𝛂_2 <𝛂_3 <…<𝛂_k ((2^(289) +1)/(2^(17) +1))=2^𝛂_1 +2^𝛂_2 +…+2^𝛂_k k=? 𝛂_1 , 𝛂_2 ,𝛂_3 ....𝛂_k positive increasing integers

Question Number 162539 Answers: 2 Comments: 0

Calculate lim_(h→0) ((f(3−h)−f(3))/(2h)), with f′(3)=2

$${Calculate}\: \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left(\mathrm{3}−{h}\right)−{f}\left(\mathrm{3}\right)}{\mathrm{2}{h}},\:{with}\:{f}'\left(\mathrm{3}\right)=\mathrm{2} \\ $$

Question Number 162535 Answers: 2 Comments: 3

prove that Ω = ∫_0 ^( ∞) (( ln ((1/x) ))/( x^( 4) + 17x^( 2) + 16)) dx=^? (π/(60)) ln(2)

$$ \\ $$$$\:\:\:\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\:\mathrm{ln}\:\left(\frac{\mathrm{1}}{{x}}\:\right)}{\:{x}^{\:\mathrm{4}} \:+\:\mathrm{17}{x}^{\:\mathrm{2}} \:+\:\mathrm{16}}\:{dx}\overset{?} {=}\:\frac{\pi}{\mathrm{60}}\:\mathrm{ln}\left(\mathrm{2}\right) \\ $$$$ \\ $$

Question Number 162530 Answers: 4 Comments: 10

Question Number 162525 Answers: 2 Comments: 0

∫_0 ^∞ ((√x)/((x^2 +4x+4)))=?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\sqrt{{x}}}{\left({x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{4}\right)}=? \\ $$

Question Number 162523 Answers: 2 Comments: 0

lim_(x→0) ((7tan x−tan 7x)/x^3 ) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7tan}\:{x}−\mathrm{tan}\:\mathrm{7}{x}}{{x}^{\mathrm{3}} }\:=? \\ $$

Question Number 162522 Answers: 1 Comments: 0

Determine all positive integers N which the sphere x^2 + y^2 + z^2 = N has an inseribed regular tetrahedron whose vertices have integer coordinates

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\boldsymbol{\mathrm{N}}\:\mathrm{which}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{N} \\ $$$$\mathrm{has}\:\mathrm{an}\:\mathrm{inseribed}\:\mathrm{regular}\:\mathrm{tetrahedron} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{have}\:\mathrm{integer}\:\mathrm{coordinates} \\ $$

Question Number 162521 Answers: 0 Comments: 0

For every positive real number x , let g(x) =lim_(r→0) ((x+1)^(r+1) - x^(r+1) )^(1/r) Find: lim_(x→∞) ((g(x))/x)

$$\mathrm{For}\:\mathrm{every}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\:\boldsymbol{\mathrm{x}}\:,\:\mathrm{let} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\underset{\boldsymbol{\mathrm{r}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\left(\mathrm{x}+\mathrm{1}\right)^{\boldsymbol{\mathrm{r}}+\mathrm{1}} \:-\:\mathrm{x}^{\boldsymbol{\mathrm{r}}+\mathrm{1}} \right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{r}}}} \\ $$$$\mathrm{Find}:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{g}\left(\mathrm{x}\right)}{\mathrm{x}} \\ $$

Question Number 162516 Answers: 0 Comments: 1

differenciate using implicit function 2x+4y+sin xy=3

$${differenciate}\:{using}\:{implicit}\:{function}\:\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{sin}\:{xy}=\mathrm{3} \\ $$

Question Number 162513 Answers: 2 Comments: 0

∫_0 ^( ∞) ((log(x))/((x+1)(x+9)))

$$\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{log}\left(\mathrm{x}\right)}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{9}\right)} \\ $$

Question Number 162512 Answers: 0 Comments: 1

solve ∫(√(cosec^2 x−2)) dx

$${solve}\:\int\sqrt{{cosec}^{\mathrm{2}} {x}−\mathrm{2}}\:{dx} \\ $$

Question Number 162506 Answers: 0 Comments: 1

Question Number 162490 Answers: 1 Comments: 0

Question Number 162496 Answers: 1 Comments: 0

Question Number 162520 Answers: 1 Comments: 0

Find: 𝛀 =∫_( 0) ^( 𝛑) (((x cos x)/(1 + sin x)))^2 dx

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\boldsymbol{\pi}} {\int}}\:\left(\frac{\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{sin}\:\mathrm{x}}\right)^{\mathrm{2}} \mathrm{dx}\: \\ $$

Question Number 162481 Answers: 1 Comments: 0

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