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Question Number 165354    Answers: 0   Comments: 0

Question Number 165349    Answers: 1   Comments: 0

Question Number 165339    Answers: 1   Comments: 0

# Advanced Calculus # Let , f : R → Q is a continuous function . prove that ” f ” is a constant function . ■ m.n ∗ Adopted from mathematical analysis book ∗ −−−−−−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:#\:{Advanced}\:\:\:{Calculus}\:#\:\:\: \\ $$$$\:\:\:\:\:\mathrm{L}{et}\:,\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{Q}\:\:{is}\:\:{a}\:\:{continuous} \\ $$$$\:\:\:\:\:\:{function}\:\:.\:{prove}\:{that}\:\:''\:{f}\:''\:{is}\:{a} \\ $$$$\:\:\:\:\:\:\:{constant}\:{function}\:.\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$\:\:\:\:\:\:\:\ast\:{Adopted}\:{from}\:{mathematical}\:{analysis}\:{book}\:\:\ast\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−−−−−−− \\ $$$$ \\ $$

Question Number 165329    Answers: 2   Comments: 0

prove ( n∈ N ) 3(n+1) ∣ n^( 3) + (n+1)^( 3) + (n+2 )^( 3)

$$ \\ $$$$\:\:\:\:{prove}\:\:\:\:\:\:\:\:\:\:\left(\:{n}\in\:\mathbb{N}\:\right) \\ $$$$\:\:\:\:\mathrm{3}\left({n}+\mathrm{1}\right)\:\mid\:{n}^{\:\mathrm{3}} \:+\:\left({n}+\mathrm{1}\right)^{\:\mathrm{3}} +\:\left({n}+\mathrm{2}\:\right)^{\:\mathrm{3}} \\ $$$$ \\ $$

Question Number 165328    Answers: 3   Comments: 0

prove that Nice Integral 𝛗=∫_0 ^( 1) (( tan^( −1) (x^( (3/2)) ))/x^( 2) ) dx =((π + (√3) ln(7 +4(√3) ))/4) ■ m.n −−−−−−−−−

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\mathscr{N}{ice}\:\:\:\mathscr{I}{ntegral} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}^{\:\frac{\mathrm{3}}{\mathrm{2}}} \right)}{{x}^{\:\mathrm{2}} }\:{dx}\:\:=\frac{\pi\:+\:\sqrt{\mathrm{3}}\:{ln}\left(\mathrm{7}\:+\mathrm{4}\sqrt{\mathrm{3}}\:\right)}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:−−−−−−−−−\:\:\: \\ $$

Question Number 165322    Answers: 1   Comments: 2

who can prove that 2^n −1produces a prime number when n is a prime number

$${who}\:{can}\:{prove}\:{that}\:\mathrm{2}^{{n}} −\mathrm{1}{produces} \\ $$$${a}\:{prime}\:{number}\:{when}\:\:{n}\:\:{is}\:{a}\: \\ $$$${prime}\:{number} \\ $$

Question Number 165321    Answers: 2   Comments: 0

Question Number 165320    Answers: 1   Comments: 0

Ω = ∫_0 ^( (π/4)) cos (2x ).e^( ⌊ sin(x)+ cos(x) ⌋) dx ⌊ x ⌋= max { m ∈ Z ∣ m ≤ x } −−−−

$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:{cos}\:\left(\mathrm{2}{x}\:\right).{e}^{\:\lfloor\:{sin}\left({x}\right)+\:{cos}\left({x}\right)\:\rfloor} {dx} \\ $$$$\:\:\:\lfloor\:{x}\:\rfloor=\:{max}\:\left\{\:{m}\:\in\:\mathbb{Z}\:\mid\:\:{m}\:\leqslant\:{x}\:\right\} \\ $$$$\:\:\:\:\:\:\:\:−−−− \\ $$

Question Number 165314    Answers: 0   Comments: 2

sin^2 1+sin^2 2+sin^2 3+.....+sin^2 90=?

$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{3}+.....+{sin}^{\mathrm{2}} \mathrm{90}=? \\ $$

Question Number 165301    Answers: 1   Comments: 0

Question Number 165296    Answers: 2   Comments: 1

Question Number 165307    Answers: 0   Comments: 0

Question Number 165291    Answers: 1   Comments: 0

Question Number 165309    Answers: 0   Comments: 0

(6/(6+(√6))) + (6/(6^2 +(√6))) +(6/(6^3 +(√6))) +...+(6/(6^(1000) +(√6))) =?

$$\:\frac{\mathrm{6}}{\mathrm{6}+\sqrt{\mathrm{6}}}\:+\:\frac{\mathrm{6}}{\mathrm{6}^{\mathrm{2}} +\sqrt{\mathrm{6}}}\:+\frac{\mathrm{6}}{\mathrm{6}^{\mathrm{3}} +\sqrt{\mathrm{6}}}\:+...+\frac{\mathrm{6}}{\mathrm{6}^{\mathrm{1000}} +\sqrt{\mathrm{6}}}\:=? \\ $$

Question Number 165262    Answers: 2   Comments: 0

find the sum Σ_(n=1) ^∞ ((1/2))^n + i ( (1/3) )^n

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{sum}}\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\boldsymbol{{n}}} \:+\:\boldsymbol{{i}}\:\left(\:\frac{\mathrm{1}}{\mathrm{3}}\:\right)^{\boldsymbol{{n}}} \: \\ $$

Question Number 165255    Answers: 1   Comments: 0

nice integral ∫_0 ^1 (1/x)ln(Σ_(m=0) ^n x^m )dx=? −−−−−−−−−−−−−by MATH.AMIN

$$\boldsymbol{\mathrm{nice}}\:\boldsymbol{\mathrm{integral}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{ln}}\left(\underset{\boldsymbol{\mathrm{m}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{m}}} \right)\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$−−−−−−−−−−−−−\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{MATH}}.\boldsymbol{\mathrm{AMIN}} \\ $$

Question Number 165243    Answers: 1   Comments: 0

⌊Find the value of x???⌋ ⌊5x + ((5x)/((5 + 5)^5 )) × (−5x) ÷ 555x −55x ×((5÷5)/5) + x = 555555555555555⌋ ^(proof:z.a)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lfloor\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{{x}}???\rfloor \\ $$$$\:\:\:\lfloor\mathrm{5}\boldsymbol{{x}}\:+\:\frac{\mathrm{5}\boldsymbol{{x}}}{\left(\mathrm{5}\:+\:\mathrm{5}\right)^{\mathrm{5}} }\:×\:\left(−\mathrm{5}\boldsymbol{{x}}\right)\:\boldsymbol{\div}\:\mathrm{555}\boldsymbol{{x}}\:−\mathrm{55}\boldsymbol{{x}}\:×\frac{\mathrm{5}\boldsymbol{\div}\mathrm{5}}{\mathrm{5}}\:\:+\:\boldsymbol{{x}}\:=\:\mathrm{555555555555555}\rfloor \\ $$$$\:^{\mathrm{proof}:\mathrm{z}.\mathrm{a}} \\ $$

Question Number 165240    Answers: 1   Comments: 0

soit la serie de fonction Σ_(n=2 ) (x^n /(nx+ln(n))) etudie la convergence simple sur [0,1[

$$\:{soit}\:{la}\:{serie}\:{de}\:{fonction}\:\underset{{n}=\mathrm{2}\:\:\:} {\sum}\frac{{x}^{{n}} }{{nx}+{ln}\left({n}\right)} \\ $$$${etudie}\:{la}\:{convergence}\:{simple}\:{sur}\:\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$

Question Number 165234    Answers: 0   Comments: 1

Question Number 165253    Answers: 3   Comments: 0

Question Number 165232    Answers: 1   Comments: 1

Question Number 165250    Answers: 0   Comments: 1

Question Number 165229    Answers: 0   Comments: 1

Σ_(i=1) ^(n=∞) lim_(x→0) ((xyn^x )/(nx_1 ))xz∫((xn^n x)/(nx^n y))dn (√(c^n +n_1 )) ∫((xn^(n!) )/(n^n xn^x ))dn

$$\underset{{i}=\mathrm{1}} {\overset{{n}=\infty} {\sum}}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{xyn}^{{x}} }{{nx}_{\mathrm{1}} }{xz}\int\frac{{xn}^{{n}} {x}}{{nx}^{{n}} {y}}{dn} \\ $$$$\sqrt{{c}^{{n}} +{n}_{\mathrm{1}} } \\ $$$$\int\frac{{xn}^{{n}!} }{{n}^{{n}} {xn}^{{x}} }{dn} \\ $$

Question Number 165226    Answers: 1   Comments: 0

make x the subject of the formula; a^x +bx+c=0

$${make}\:{x}\:{the}\:{subject}\:{of}\:{the}\:{formula}; \\ $$$${a}^{{x}} +{bx}+{c}=\mathrm{0} \\ $$

Question Number 165217    Answers: 2   Comments: 5

Question Number 165218    Answers: 1   Comments: 1

Question by M.N July Φ = ∫_0 ^( 1) ((ln(1 + x^4 + x^8 ))/x)dx Φ =^(x=x^(1/2) ) (1/2)∫_0 ^( 1) ((ln(1+x^2 +x^4 ))/x)dx Φ = (1/2)∫_0 ^( 1) ((ln(((1−x^2 )/(1−x^6 ))))/x)dx = (1/2)∫_0 ^( 1) ((ln(1−x^2 ))/x)dx − (1/2)∫_0 ^( 1) ((ln(1−x^6 ))/x)dx Φ = (1/2)(A − B) A =^(x=x^(1/2) ) (1/2)∫_0 ^( 1) ((ln(1−x))/x)dx = (1/2)Li_2 (1) B =^(x=x^(1/6) ) (1/6)∫_0 ^( 1) ((x^((1/6)−1) ln(1−x))/x^(1/6) )dx B = (1/6)∫_0 ^( 1) ((ln(1−x))/x)dx = (1/6)Li_2 (1) Φ = (1/2)((1/2)Li_2 (1)−(1/6)Li_2 (1)) = (1/6)Li_2 (1) 𝚽 = ((𝛇(2))/3) ▲▲▲

$$\mathrm{Question}\:\mathrm{by}\:\boldsymbol{\mathrm{M}}.\boldsymbol{\mathrm{N}}\:\boldsymbol{\mathrm{July}} \\ $$$$\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{x}^{\mathrm{8}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$\Phi\:\overset{\mathrm{x}=\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} } {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$\Phi\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\frac{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }{\mathrm{1}−\mathrm{x}^{\mathrm{6}} }\right)}{\mathrm{x}}\mathrm{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}}\mathrm{dx}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{6}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$\Phi\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{A}\:−\:\mathrm{B}\right) \\ $$$$\mathrm{A}\:\overset{\mathrm{x}=\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} } {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\mathrm{1}\right) \\ $$$$\mathrm{B}\:\overset{\mathrm{x}=\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} } {=}\frac{\mathrm{1}}{\mathrm{6}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}−\mathrm{1}} \mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} }\mathrm{dx} \\ $$$$\mathrm{B}\:=\:\frac{\mathrm{1}}{\mathrm{6}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}\:=\:\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\mathrm{1}\right)\: \\ $$$$\Phi\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\mathrm{1}\right)−\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\mathrm{1}\right)\right)\:=\:\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\mathrm{1}\right) \\ $$$$\boldsymbol{\Phi}\:=\:\frac{\boldsymbol{\zeta}\left(\mathrm{2}\right)}{\mathrm{3}}\:\blacktriangle\blacktriangle\blacktriangle \\ $$

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