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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

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 Ω = Σ_(n=1) ^∞ (( (−1)^( n) n)/(3^( n) (2n −1 ))) =? − Inspired from Sir Ghaderi′s post−

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

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) ) ≥ (π/2)

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

Question Number 162414 Answers: 1 Comments: 0

Prove the Identity for any (a,n) in Real Number (1 + a)∙a^([n]) = a ∙ a^(2[(n/2)]) + a^(2[((n+1)/2)]) [∗] Greatest Integer Function

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{Identity}\:\mathrm{for}\:\mathrm{any}\:\left(\mathrm{a},\mathrm{n}\right)\:\mathrm{in}\:\mathrm{Real}\:\mathrm{Number} \\ $$$$\left(\mathrm{1}\:+\:\mathrm{a}\right)\centerdot\mathrm{a}^{\left[\boldsymbol{\mathrm{n}}\right]} \:=\:\mathrm{a}\:\centerdot\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}}\right]} \:+\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}+\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$

Question Number 162411 Answers: 0 Comments: 0

Prove the identity for any ′n′ in Real number [(n/2)] ∙ [((n + 1)/2)] = (1/4)([n]^2 + 2[(n/2)] - [n]) [∗] Greatest Integer Function

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{for}\:\mathrm{any}\:'\boldsymbol{\mathrm{n}}'\:\mathrm{in}\:\mathrm{Real}\:\mathrm{number} \\ $$$$\left[\frac{\mathrm{n}}{\mathrm{2}}\right]\:\centerdot\:\left[\frac{\mathrm{n}\:+\:\mathrm{1}}{\mathrm{2}}\right]\:=\:\frac{\mathrm{1}}{\mathrm{4}}\left(\left[\mathrm{n}\right]^{\mathrm{2}} \:+\:\mathrm{2}\left[\frac{\mathrm{n}}{\mathrm{2}}\right]\:-\:\left[\mathrm{n}\right]\right) \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$

Question Number 162410 Answers: 1 Comments: 0

∫(dx/((a−cosx)^2 )) a>1

$$\int\frac{{dx}}{\left({a}−{cosx}\right)^{\mathrm{2}} }\:\:\:{a}>\mathrm{1} \\ $$

Question Number 162510 Answers: 1 Comments: 0

lim_(x→0) ((7tan x−tan 7x)/(3x)) =?

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

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