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∫ x^dx

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1. Let f(z)= { (( 1/q z=p/q , p,q∈Z , gcd(p,q)=1 , q>0)),((0 z∈R\Q)) :} ∫_( R) f(z)dz=? 2. Show that f(z) is continous function when z∈R\Q 3.Show that f(z) is NOT continous when z∈Q

$$\mathrm{1}.\:\mathrm{Let}\:{f}\left({z}\right)=\begin{cases}{\:\mathrm{1}/{q}\:\:\:\:\:{z}={p}/{q}\:,\:{p},{q}\in\mathbb{Z}\:,\:\mathrm{gcd}\left({p},{q}\right)=\mathrm{1}\:,\:{q}>\mathrm{0}}\\{\mathrm{0}\:\:\:\:{z}\in\mathbb{R}\backslash\mathbb{Q}}\end{cases} \\ $$$$\int_{\:\mathbb{R}} \:{f}\left({z}\right)\mathrm{d}{z}=? \\ $$$$\mathrm{2}.\:\mathrm{Show}\:\mathrm{that}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{continous}\:\mathrm{function}\:\mathrm{when}\:{z}\in\mathbb{R}\backslash\mathbb{Q} \\ $$$$\mathrm{3}.\mathrm{Show}\:\mathrm{that}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{NOT}\:\mathrm{continous}\:\mathrm{when}\:{z}\in\mathbb{Q} \\ $$

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x^2 +x+1=1+(1/(1+(1/(1+(1/(1+(1/(...)))))))) x=?

$${x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{...}}}} \\ $$$${x}=? \\ $$

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3^(444) + 4^(333) Find the remainder when dividing the number by 7

$$\mathrm{3}^{\mathrm{444}} \:\:+\:\:\mathrm{4}^{\mathrm{333}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{dividing}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{by}\:\mathrm{7} \\ $$

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((16∙3^(x−1) + 16∙3^x )/(4∙3^(x+1) − 2∙3^(x−1) )) = ((2^(x−1) )^(1/3) /(17)) find: x=?

$$\frac{\mathrm{16}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{x}}−\mathrm{1}} \:+\:\mathrm{16}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{x}}+\mathrm{1}} \:−\:\mathrm{2}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{x}}−\mathrm{1}} }\:=\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{2}^{\boldsymbol{\mathrm{x}}−\mathrm{1}} }}{\mathrm{17}}\:\:\:\:\:\mathrm{find}:\:\:\mathrm{x}=? \\ $$

Question Number 227369 Answers: 1 Comments: 0

{ ((3^m + 2∙4^(n+1) = 17)),((4^n − 5∙3^m = −44)) :} find: m+n=?

$$\begin{cases}{\mathrm{3}^{\boldsymbol{\mathrm{m}}} \:+\:\mathrm{2}\centerdot\mathrm{4}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:=\:\mathrm{17}}\\{\mathrm{4}^{\boldsymbol{\mathrm{n}}} \:−\:\mathrm{5}\centerdot\mathrm{3}^{\boldsymbol{\mathrm{m}}} \:=\:−\mathrm{44}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\mathrm{m}+\mathrm{n}=? \\ $$

Question Number 227368 Answers: 2 Comments: 0

If sin9° = a Find ((cos3°)/(sin15°)) − ((sin3°)/(cos15°)) = ?

$$\mathrm{If}\:\:\:\mathrm{sin9}°\:=\:\mathrm{a} \\ $$$$\mathrm{Find}\:\:\:\frac{\mathrm{cos3}°}{\mathrm{sin15}°}\:−\:\frac{\mathrm{sin3}°}{\mathrm{cos15}°}\:=\:? \\ $$

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Prove that in any acute △ABC if I is the in-center and H is the ortho-center then: (1/(IA)) + (1/(IB)) + (1/(IC)) ≤ (1/(HA)) + (1/(HB)) + (1/(HC))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\mathrm{acute}\:\bigtriangleup\mathrm{ABC}\: \\ $$$$\mathrm{if}\:\mathrm{I}\:\mathrm{is}\:\mathrm{the}\:\mathrm{in}-\mathrm{center}\:\mathrm{and}\:\mathrm{H}\:\mathrm{is}\:\mathrm{the}\:\mathrm{ortho}-\mathrm{center} \\ $$$$\mathrm{then}: \\ $$$$\frac{\mathrm{1}}{\mathrm{IA}}\:+\:\frac{\mathrm{1}}{\mathrm{IB}}\:+\:\frac{\mathrm{1}}{\mathrm{IC}}\:\:\leqslant\:\:\frac{\mathrm{1}}{\mathrm{HA}}\:+\:\frac{\mathrm{1}}{\mathrm{HB}}\:+\:\frac{\mathrm{1}}{\mathrm{HC}} \\ $$

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