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Question Number 164890 Answers: 0 Comments: 0

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Solve for natural numbers: 𝛟(n) ∙ 𝛟(n + 1) = 288n 𝛟-Euler′s totient function

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{natural}\:\mathrm{numbers}: \\ $$$$\boldsymbol{\varphi}\left(\boldsymbol{\mathrm{n}}\right)\:\centerdot\:\boldsymbol{\varphi}\left(\boldsymbol{\mathrm{n}}\:+\:\mathrm{1}\right)\:=\:\mathrm{288}\boldsymbol{\mathrm{n}} \\ $$$$\boldsymbol{\varphi}-\mathrm{Euler}'\mathrm{s}\:\mathrm{totient}\:\mathrm{function} \\ $$

Question Number 164874 Answers: 1 Comments: 0

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Let a,b,c >0; 42abc + 4bca +1cab ≤ 24 Then (1/(ab)) + (2/(bc)) + (3/(ca)) = ?? ^({Z.A})

$$\:\:\:\:\boldsymbol{\mathrm{Let}}\:\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}\:>\mathrm{0}; \\ $$$$\:\:\:\:\mathrm{42}\boldsymbol{{abc}}\:+\:\mathrm{4}\boldsymbol{{bca}}\:+\mathrm{1}\boldsymbol{{cab}}\:\leqslant\:\mathrm{24} \\ $$$$\:\:\:\:\mathrm{T}\boldsymbol{\mathrm{hen}}\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{ab}}}\:+\:\frac{\mathrm{2}}{\boldsymbol{\mathrm{bc}}}\:+\:\frac{\mathrm{3}}{\boldsymbol{\mathrm{ca}}}\:=\:?? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:^{\left\{{Z}.\mathrm{A}\right\}} \\ $$

Question Number 164854 Answers: 1 Comments: 0

Evaluate the Integral; [∫_0 ^∞ ((x^2 − 1)/(1−x)) dx =??] ^({Z.A})

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Evaluate}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{Integral}}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\int_{\mathrm{0}} ^{\infty} \:\frac{\boldsymbol{{x}}^{\mathrm{2}} \:−\:\mathrm{1}}{\mathrm{1}−\boldsymbol{{x}}}\:\boldsymbol{{dx}}\:=??\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:^{\left\{\boldsymbol{{Z}}.\mathrm{A}\right\}} \\ $$

Question Number 164853 Answers: 0 Comments: 0

Ω = ∫_0 ^( 1) (( ln(1+x ).ln(x))/(1−x)) dx

$$ \\ $$$$\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\left(\mathrm{1}+{x}\:\right).{ln}\left({x}\right)}{\mathrm{1}−{x}}\:{dx} \\ $$

Question Number 164842 Answers: 3 Comments: 13

Question Number 164826 Answers: 2 Comments: 0

if h(x) = x − ⌊(1/x)⌋ , x>0 then h^( −1) ( x ) = ?

$$ \\ $$$$\:\:\:\:\:\:{if}\:\:\:\:{h}\left({x}\right)\:=\:{x}\:−\:\lfloor\frac{\mathrm{1}}{{x}}\rfloor\:\:,\:\:{x}>\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:{then}\:\:\:{h}^{\:−\mathrm{1}} \left(\:{x}\:\right)\:=\:? \\ $$$$ \\ $$

Question Number 164821 Answers: 0 Comments: 9

prove that n=((360^° )/∝)−1 n=number of multiple images

$${prove}\:{that}\:\:{n}=\frac{\mathrm{360}^{°} }{\propto}−\mathrm{1} \\ $$$${n}={number}\:{of}\:\:{multiple}\:{images}\: \\ $$

Question Number 164820 Answers: 1 Comments: 0

Question Number 164814 Answers: 0 Comments: 2

does the forum support animated images in gif format?

$${does}\:{the}\:{forum}\:{support}\:{animated} \\ $$$${images}\:{in}\:{gif}\:{format}? \\ $$

Question Number 164810 Answers: 1 Comments: 2

find n^(th) terms of <1,0,1,0,1,0,....>

$${find}\:{n}^{{th}} \:{terms}\:{of}\:<\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{0},....> \\ $$

Question Number 164809 Answers: 2 Comments: 0

1) ∫((x)^(1/3) /( (√x)+(x)^(1/4) ))=? 2) ∫(x/((x^2 +2x+2)^2 ))=?

$$\left.\mathrm{1}\right)\:\:\:\:\int\frac{\sqrt[{\mathrm{3}}]{{x}}}{\:\sqrt{{x}}+\sqrt[{\mathrm{4}}]{{x}}}=? \\ $$$$\left.\mathrm{2}\right)\:\:\:\:\int\frac{{x}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{\mathrm{2}} }=? \\ $$

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Question Number 164788 Answers: 1 Comments: 0

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Question Number 164770 Answers: 2 Comments: 0

{ ((f(3x−1)+g(6x−1)=3x)),((f(x+1)+x g(2x+3)=2x^2 +x)) :} f(x)=?

$$\begin{cases}{{f}\left(\mathrm{3}{x}−\mathrm{1}\right)+{g}\left(\mathrm{6}{x}−\mathrm{1}\right)=\mathrm{3}{x}}\\{{f}\left({x}+\mathrm{1}\right)+{x}\:{g}\left(\mathrm{2}{x}+\mathrm{3}\right)=\mathrm{2}{x}^{\mathrm{2}} +{x}}\end{cases} \\ $$$$\:{f}\left({x}\right)=? \\ $$

Question Number 164760 Answers: 2 Comments: 0

tan193=k cos167=?

$${tan}\mathrm{193}={k} \\ $$$${cos}\mathrm{167}=? \\ $$

Question Number 164758 Answers: 1 Comments: 0

tan22=(2/5) ((sin112+sin202)/(cos(−68)))=?

$${tan}\mathrm{22}=\frac{\mathrm{2}}{\mathrm{5}} \\ $$$$\frac{{sin}\mathrm{112}+{sin}\mathrm{202}}{{cos}\left(−\mathrm{68}\right)}=? \\ $$

Question Number 164757 Answers: 0 Comments: 1

cos α=(2/3) sin((π/2)+α)−3sin^2 (((7π)/2)+α)−2cos(5π+α)=?

$$\mathrm{cos}\:\alpha=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${sin}\left(\frac{\pi}{\mathrm{2}}+\alpha\right)−\mathrm{3}{sin}^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{2}}+\alpha\right)−\mathrm{2}{cos}\left(\mathrm{5}\pi+\alpha\right)=? \\ $$

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