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Question Number 164134 Answers: 1 Comments: 0

Question Number 164133 Answers: 1 Comments: 0

Question Number 164129 Answers: 1 Comments: 0

prove 𝛗= Re (∫_0 ^( 1) Li_( 2) ( (1/x) ) )dx = ζ (2) −−−m.n−−−

$$ \\ $$$$\:\:\:{prove} \\ $$$$ \\ $$$$\:\:\:\:\:\boldsymbol{\phi}=\:\mathscr{R}{e}\:\left(\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{Li}_{\:\mathrm{2}} \:\left(\:\frac{\mathrm{1}}{{x}}\:\right)\:\right){dx}\:=\:\zeta\:\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:−−−{m}.{n}−−− \\ $$

Question Number 164125 Answers: 1 Comments: 0

Prove that: ∫_( 0) ^( (𝛑/2)) ∫_( 0) ^( 1) arctan (((sinx)/(u + cosx))) dudx = (π^2 /(16)) + (3/2) ln(2) - (π/4)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{arctan}\:\left(\frac{\mathrm{sin}\boldsymbol{\mathrm{x}}}{\mathrm{u}\:+\:\mathrm{cos}\boldsymbol{\mathrm{x}}}\right)\:\mathrm{dudx}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{16}}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{ln}\left(\mathrm{2}\right)\:-\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 164124 Answers: 1 Comments: 0

Solve for real numbers: ((16^x + 4^x + 1^x )/(4^x + 2^x + 1^x )) = ((8^x + 1)/(65))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{16}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}^{\boldsymbol{\mathrm{x}}} }\:=\:\frac{\mathrm{8}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}}{\mathrm{65}} \\ $$

Question Number 164123 Answers: 0 Comments: 0

very nice to problem: find in closed form; ∫_0 ^1 log (1−x^2 ) log^(n ) (1−x) dx; n ∈ N^+ ^(z.)

$$\mathrm{very}\:\mathrm{nice}\:\mathrm{to}\:\mathrm{problem}: \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{in}}\:\boldsymbol{{closed}}\:\boldsymbol{{form}}; \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{{log}}\:\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} \right)\:\boldsymbol{{log}}\:^{\boldsymbol{{n}}\:} \:\left(\mathrm{1}−\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{n}}\:\in\:\:\mathbb{N}^{+} \\ $$$$\:^{\mathrm{z}.} \\ $$

Question Number 164122 Answers: 1 Comments: 0

lim_(n→∞) Σ_(k=1) ^n sin (1/(n+k))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}=? \\ $$

Question Number 164121 Answers: 1 Comments: 3

if w = f(x,y) and x = r cosθ , y = rsinθ then prove that w_(rr) + w_(θθ) = 0?

$${if}\:{w}\:=\:{f}\left({x},{y}\right)\:{and}\:{x}\:=\:{r}\:{cos}\theta\:,\:{y}\:=\:{rsin}\theta \\ $$$$ \\ $$$${then}\:{prove}\:{that}\:{w}_{{rr}} \:+\:{w}_{\theta\theta} \:=\:\mathrm{0}? \\ $$

Question Number 164120 Answers: 0 Comments: 1

How do you all to prove Integral; Prove the; ∫ (((In x)2)/x) dx = (1/3) (In x)^3

$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{all}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{Integral}; \\ $$$$\:\boldsymbol{{Prove}}\:\boldsymbol{{the}}; \\ $$$$\:\int\:\frac{\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)\mathrm{2}}{\boldsymbol{{x}}}\:\boldsymbol{{dx}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)^{\mathrm{3}} \\ $$

Question Number 164117 Answers: 0 Comments: 0

soit (a_n )_n une suite define par a_n =Σ_(k=1) ^n (1/k)−ln(n) montrons a_n converge

$${soit}\:\left({a}_{{n}} \right)_{{n}} {une}\:{suite}\:{define}\:{par} \\ $$$${a}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}−{ln}\left({n}\right)\:\:{montrons}\: \\ $$$${a}_{{n}} {converge} \\ $$

Question Number 164116 Answers: 0 Comments: 1

(a+b+c+x)^(100) 50^(th) limit is equl to=?

$$\left({a}+{b}+{c}+{x}\right)^{\mathrm{100}} \\ $$$$\mathrm{50}^{{th}} \:{limit}\:{is}\:{equl}\:{to}=? \\ $$

Question Number 164118 Answers: 1 Comments: 0

What is Mathematical Calculus and Mathematical fhysics, Then what is the difference and how to explain?? ^([Zaynal])

$$\boldsymbol{{What}}\:\boldsymbol{{is}} \\ $$$$\:\boldsymbol{{Mathematical}}\:\boldsymbol{{Calculus}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{Mathematical}}\:\boldsymbol{{fhysics}}, \\ $$$$\boldsymbol{{Then}}\:\boldsymbol{{what}}\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{difference}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{how}}\:\boldsymbol{{to}}\:\boldsymbol{{explain}}?? \\ $$$$\:^{\left[\mathrm{Zaynal}\right]} \\ $$

Question Number 164103 Answers: 1 Comments: 0

prove that Ω=∫_0 ^( 1) ln(((1+x)/(1−x)) ).(dx/(x (√( 1−x^( 2) )))) = (π^( 2) /2) −− m.n−−

$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\:\right).\frac{{dx}}{{x}\:\sqrt{\:\mathrm{1}−{x}^{\:\mathrm{2}} }}\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{2}} \\ $$$$\:\:\:\:\:−−\:{m}.{n}−− \\ $$$$ \\ $$

Question Number 164468 Answers: 2 Comments: 0

Question Number 164466 Answers: 0 Comments: 1

Question Number 164085 Answers: 0 Comments: 0

Question Number 164077 Answers: 1 Comments: 0

Question Number 164073 Answers: 2 Comments: 1

Question Number 164071 Answers: 1 Comments: 0

Question Number 164068 Answers: 1 Comments: 0

Question Number 164059 Answers: 0 Comments: 0

Air leaks from a spherical ballon so that it maintains its shape at a rate of 25 cc/m .What is the rate of change in the length of the radius of the balloon when the radius is 5 cm

$$\:\:\mathrm{Air}\:\mathrm{leaks}\:\mathrm{from}\:\mathrm{a}\:\mathrm{spherical}\:\mathrm{ballon}\:\mathrm{so}\:\mathrm{that}\: \\ $$$$\:\mathrm{it}\:\mathrm{maintains}\:\mathrm{its}\:\mathrm{shape}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{25}\:\mathrm{cc}/\mathrm{m} \\ $$$$\:.\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{change}\:\mathrm{in}\:\mathrm{the}\:\mathrm{length} \\ $$$$\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon}\:\mathrm{when}\:\mathrm{the}\:\mathrm{radius} \\ $$$$\:\:\mathrm{is}\:\mathrm{5}\:\mathrm{cm} \\ $$

Question Number 164052 Answers: 0 Comments: 0

Question Number 164050 Answers: 1 Comments: 0

Question Number 164039 Answers: 1 Comments: 0

consider f function Df=[0,1] f(0)=f(1) c∈[0,(1/2)] show that f(c)=f(c+(1/2))

$${consider}\:{f}\:{function}\:{Df}=\left[\mathrm{0},\mathrm{1}\right] \\ $$$${f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right)\:{c}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right]\:{show}\:{that}\:{f}\left({c}\right)={f}\left({c}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 164027 Answers: 1 Comments: 2

Question Number 164024 Answers: 1 Comments: 0

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