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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

$$\boldsymbol{\alpha}_{\mathrm{1}} <\boldsymbol{\alpha}_{\mathrm{2}} <\boldsymbol{\alpha}_{\mathrm{3}} <\ldots<\boldsymbol{\alpha}_{{k}} \\ $$$$\frac{\mathrm{2}^{\mathrm{289}} +\mathrm{1}}{\mathrm{2}^{\mathrm{17}} +\mathrm{1}}=\mathrm{2}^{\boldsymbol{\alpha}_{\mathrm{1}} } +\mathrm{2}^{\boldsymbol{\alpha}_{\mathrm{2}} } +\ldots+\mathrm{2}^{\boldsymbol{\alpha}_{{k}} } \:\:\:\:\:\:\:\boldsymbol{\mathrm{k}}=? \\ $$$$ \\ $$$$\boldsymbol{\alpha}_{\mathrm{1}} ,\:\boldsymbol{\alpha}_{\mathrm{2}} ,\boldsymbol{\alpha}_{\mathrm{3}} ....\boldsymbol{\alpha}_{{k}} \\ $$positive increasing integers

Question Number 162539    Answers: 2   Comments: 0

Calculate lim_(hβ†’0) ((f(3βˆ’h)βˆ’f(3))/(2h)), with fβ€²(3)=2

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

Question Number 162535    Answers: 2   Comments: 3

prove that Ξ© = ∫_0 ^( ∞) (( ln ((1/x) ))/( x^( 4) + 17x^( 2) + 16)) dx=^? (Ο€/(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_(xβ†’0) ((7tan xβˆ’tan 7x)/x^3 ) =?

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

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_(rβ†’0) ((x+1)^(r+1) - x^(r+1) )^(1/r) Find: lim_(xβ†’βˆž) ((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 xβˆ’2)) dx

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

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

Question Number 162478    Answers: 1   Comments: 0

Calculate: Σ_(k=1) ^∞ ((H_k 2^(-k) )/(k + 1)) where H_k is the k-th harmonic number

$$\mathrm{Calculate}:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{2}^{-\boldsymbol{\mathrm{k}}} }{\mathrm{k}\:+\:\mathrm{1}} \\ $$$$\mathrm{where}\:\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{k}}-\mathrm{th}\:\mathrm{harmonic}\:\mathrm{number} \\ $$

Question Number 162473    Answers: 1   Comments: 0

Question Number 162471    Answers: 2   Comments: 0

[reposted] find ∫_( 0) ^( (𝛑/2)) sin^8 (x)dx + ∫_( 0) ^( 1) sin^(-1) ((x)^(1/8) ) dx=?

$$\left[{reposted}\right] \\ $$$${find}\:\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right){dx}\:+\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{sin}^{-\mathrm{1}} \left(\sqrt[{\mathrm{8}}]{\mathrm{x}}\right)\:{dx}=? \\ $$

Question Number 162429    Answers: 0   Comments: 1

put the digits 0,1,2,3,4,5,6,7,8,9,in place of the letters in order to perform the edditon

$${put}\:{the}\:{digits}\:\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},{in}\:{place}\:{of}\:{the}\:{letters}\:{in}\:{order}\:{to}\:{perform}\:{the}\:{edditon} \\ $$

Question Number 162424    Answers: 1   Comments: 0

calculate Ξ© = Ξ£_(n=1) ^∞ (( (βˆ’1)^( n) n)/(3^( n) (2n βˆ’1 ))) =? βˆ’ Inspired from Sir Ghaderiβ€²s postβˆ’

$$ \\ $$$$\:\:\:\:\:{calculate}\: \\ $$$$ \\ $$$$\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\left(βˆ’\mathrm{1}\right)^{\:{n}} {n}}{\mathrm{3}^{\:{n}} \:\left(\mathrm{2}{n}\:βˆ’\mathrm{1}\:\right)}\:=?\:\:\:\: \\ $$$$\:\:\:\:βˆ’\:\mathrm{I}{nspired}\:{from}\:{Sir}\:\mathrm{G}{haderi}'{s}\:{post}βˆ’ \\ $$

Question Number 162533    Answers: 5   Comments: 0

Question Number 162416    Answers: 1   Comments: 1

Prove that: ∫_( 0) ^( (𝛑/4)) ((4 ln (cotx))/(cos(2x + 2022𝛑))) dx = 3𝛇(2)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\:\frac{\mathrm{4}\:\mathrm{ln}\:\left(\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}{\mathrm{cos}\left(\mathrm{2x}\:+\:\mathrm{2022}\boldsymbol{\pi}\right)}\:\mathrm{dx}\:=\:\mathrm{3}\boldsymbol{\zeta}\left(\mathrm{2}\right) \\ $$

Question Number 162417    Answers: 0   Comments: 4

Prove ∫_( 0) ^( (𝛑/2)) sin^8 (x) + ∫_( 0) ^( 1) sin^(-1) ((x)^(1/8) ) β‰₯ (Ο€/2)

$$\mathrm{Prove} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right)\:+\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{sin}^{-\mathrm{1}} \:\left(\sqrt[{\mathrm{8}}]{\mathrm{x}}\right)\:\geqslant\:\frac{\pi}{\mathrm{2}} \\ $$

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