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Question Number 212669 Answers: 1 Comments: 1

Question Number 212668 Answers: 4 Comments: 0

Question Number 212654 Answers: 1 Comments: 0

Question Number 212651 Answers: 1 Comments: 0

Question Number 212646 Answers: 1 Comments: 1

lim_(n→∞) Σ_(i=1) ^n Σ_(j=i) ^i ((i(i+j))/((n^2 +i^2 )(n^2 +j^2 )))

$$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\underset{{j}={i}} {\overset{{i}} {\sum}}\frac{{i}\left({i}+{j}\right)}{\left({n}^{\mathrm{2}} +{i}^{\mathrm{2}} \right)\left({n}^{\mathrm{2}} +{j}^{\mathrm{2}} \right)} \\ $$

Question Number 212645 Answers: 0 Comments: 0

{ ((x=2+ log _2 log _2 y)),((y=2 log _2 z )),((z=2+ log _2 log _2 x )) :}

$$\:\:\begin{cases}{\mathrm{x}=\mathrm{2}+\:\mathrm{log}\:_{\mathrm{2}} \mathrm{log}\:_{\mathrm{2}} \mathrm{y}}\\{\mathrm{y}=\mathrm{2}\:\mathrm{log}\:_{\mathrm{2}} \mathrm{z}\:}\\{\mathrm{z}=\mathrm{2}+\:\mathrm{log}\:_{\mathrm{2}} \:\mathrm{log}\:_{\mathrm{2}} \mathrm{x}\:}\end{cases} \\ $$

Question Number 212643 Answers: 0 Comments: 6

Guess we can make youtube educational videos using this forum′s editor in offline mode. Tinkutara team, let me know. I alresdy made one..here

$${Guess}\:{we}\:{can}\:{make}\:{youtube}\: \\ $$$${educational}\:{videos}\:{using}\:{this}\: \\ $$$${forum}'{s}\:{editor}\:{in}\:{offline}\:{mode}. \\ $$$${Tinkutara}\:{team},\:{let}\:{me}\:{know}. \\ $$$${I}\:{alresdy}\:{made}\:{one}..{here} \\ $$

Question Number 212635 Answers: 1 Comments: 1

m≤∫_1 ^3 ((1 )/(√(2x^2 +7 ))) .dx≤k find the value of the constant m and k

$$\:\:\boldsymbol{{m}}\leqslant\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}\:}{\sqrt{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{7}\:}}\:.\boldsymbol{{dx}}\leqslant\boldsymbol{{k}}\:\boldsymbol{{find}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{constant}} \\ $$$$\:\:\boldsymbol{{m}}\:\boldsymbol{{and}}\:\boldsymbol{{k}} \\ $$

Question Number 212630 Answers: 2 Comments: 1

Question Number 212627 Answers: 1 Comments: 0

lim_(n→∞) (((√(1∙2))/(n^2 +1))+((√(2∙3))/(n^2 +2))+…+((√(n(n+1)))/(n^2 +n)))

$$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{1}\centerdot\mathrm{2}}}{{n}^{\mathrm{2}} +\mathrm{1}}+\frac{\sqrt{\mathrm{2}\centerdot\mathrm{3}}}{{n}^{\mathrm{2}} +\mathrm{2}}+\ldots+\frac{\sqrt{{n}\left({n}+\mathrm{1}\right)}}{{n}^{\mathrm{2}} +{n}}\right) \\ $$

Question Number 212626 Answers: 1 Comments: 0

let f(x)=(1/( (√((x−a)(x−b)(x−c))))) let a, b, c ∈R ∧a<b<c ⇒ D(f(x))=(a, b)∪(c, ∞) prove ∫_a ^b f(x)dx=∫_c ^∞ f(x)dx

$$\mathrm{let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\left({x}−{a}\right)\left({x}−{b}\right)\left({x}−{c}\right)}} \\ $$$$\mathrm{let}\:{a},\:{b},\:{c}\:\in\mathbb{R}\:\wedge{a}<{b}<{c} \\ $$$$\Rightarrow\:{D}\left({f}\left({x}\right)\right)=\left({a},\:{b}\right)\cup\left({c},\:\infty\right) \\ $$$$\mathrm{prove}\:\underset{{a}} {\overset{{b}} {\int}}{f}\left({x}\right){dx}=\underset{{c}} {\overset{\infty} {\int}}{f}\left({x}\right){dx} \\ $$

Question Number 212621 Answers: 0 Comments: 1

A group of n people are standing in a circle. They are numbered with 1, 2, 3, etc. Starting with the person with the number 1, every second person is removed from the circle and this process continues until only one person remains in the circle. What will be the number of the last person left.

$$\mathrm{A}\:\mathrm{group}\:\mathrm{of}\:\boldsymbol{\mathrm{n}}\:\mathrm{people}\:\mathrm{are}\:\mathrm{standing}\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{circle}.\:\:\mathrm{They}\:\mathrm{are}\:\mathrm{numbered}\:\mathrm{with}\: \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{etc}.\:\mathrm{Starting}\:\mathrm{with}\:\mathrm{the}\:\mathrm{person} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{number}\:\mathrm{1},\:\mathrm{every}\:\mathrm{second}\: \\ $$$$\mathrm{person}\:\mathrm{is}\:\mathrm{removed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{and}\: \\ $$$$\mathrm{this}\:\mathrm{process}\:\mathrm{continues}\:\mathrm{until}\:\mathrm{only}\:\mathrm{one}\: \\ $$$$\mathrm{person}\:\mathrm{remains}\:\mathrm{in}\:\mathrm{the}\:\mathrm{circle}.\:\:\mathrm{What} \\ $$$$\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{last}\:\mathrm{person} \\ $$$$\mathrm{left}. \\ $$

Question Number 212618 Answers: 0 Comments: 1

lim_(n→0) (Π_(k=1) ^n ((k^2 +3k+2)/(k^2 +2k+1)))^(1/n)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{k}^{\mathrm{2}} +\mathrm{3}{k}+\mathrm{2}}{{k}^{\mathrm{2}} +\mathrm{2}{k}+\mathrm{1}}\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 212616 Answers: 0 Comments: 1

if f(x)=^a x=x^x^x^⋰^x (there are a x′s;a∈N and a≠0) ∫f(x)dx=?

$$ \\ $$$$ \\ $$$${if}\:\:\mathrm{f}\left(\mathrm{x}\right)=\:^{{a}} {x}={x}^{{x}^{{x}^{\iddots^{{x}} } } } \left({there}\:{are}\:{a}\:{x}'{s};{a}\in\mathbb{N}\:{and}\:{a}\neq\mathrm{0}\right) \\ $$$$\int{f}\left({x}\right){dx}=? \\ $$

Question Number 212612 Answers: 1 Comments: 0

Solve the differential equation: (dy/dx) = cos(x + y)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}\:\:=\:\:\mathrm{cos}\left(\mathrm{x}\:+\:\mathrm{y}\right) \\ $$

Question Number 212609 Answers: 1 Comments: 0

If 0≤x≤2 Prove that: 2^x + 1 − (√(10,5x + 4)) ≤ 0

$$\mathrm{If}\:\:\:\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}\:−\:\sqrt{\mathrm{10},\mathrm{5x}\:+\:\mathrm{4}}\:\leqslant\:\mathrm{0} \\ $$

Question Number 212607 Answers: 0 Comments: 0

Question Number 212604 Answers: 0 Comments: 0

draw the structural formula for Spiro [2.2.2.5.8.6] compounds

$$\boldsymbol{\mathrm{draw}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{structural}}\:\boldsymbol{\mathrm{formula}}\:\boldsymbol{\mathrm{for}} \\ $$$$ \\ $$Spiro [2.2.2.5.8.6] compounds

Question Number 212602 Answers: 0 Comments: 0

Question Number 212601 Answers: 0 Comments: 0

Question Number 212600 Answers: 1 Comments: 0

certificate: lim_(n→∞) ∫_0 ^1 (n/(1+n^2 x^2 ))e^x^3 dx=(π/2).

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{n}}{\mathrm{1}+{n}^{\mathrm{2}} {x}^{\mathrm{2}} }{e}^{{x}^{\mathrm{3}} } {dx}=\frac{\pi}{\mathrm{2}}. \\ $$

Question Number 212598 Answers: 1 Comments: 0

Help me to solve pls Q 212576

$$\mathrm{Help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{pls} \\ $$$$\mathrm{Q}\:\mathrm{212576}\: \\ $$

Question Number 212647 Answers: 2 Comments: 1

(√(x−(1/x))) +(√(1−(1/x))) = x

$$\:\:\:\sqrt{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}}\:+\sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}}\:=\:\mathrm{x}\: \\ $$

Question Number 212906 Answers: 0 Comments: 1

Pls i need a help.. from Ordinary differantial Equation t^2 y^((2)) (t)+t∙y^((1)) (t)+(t^2 −ν^2 )y(t)=0 we Already Know Solution y(t)=C_1 J_ν (t)+C_2 J_(−ν) (t) But J_(−ν) (t) can′t Satisfy as Solution Cus J_ν (t) and J_(−ν) (t) are Not Linear independent. Wronskian W∈mat(m,m) det W=0 thus Solution y(t)=C_1 J_ν (t)+C_2 Y_ν (t) i already undertand above indentity i wrote my question is prove Abel′s identity W(J_ν (t),Y_ν (t))=(2/(πt)) Pls Help :(

$$\mathrm{Pls}\:\mathrm{i}\:\mathrm{need}\:\mathrm{a}\:\mathrm{help}.. \\ $$$$\mathrm{from}\:\mathrm{Ordinary}\:\mathrm{differantial}\:\mathrm{Equation} \\ $$$${t}^{\mathrm{2}} {y}^{\left(\mathrm{2}\right)} \left({t}\right)+{t}\centerdot{y}^{\left(\mathrm{1}\right)} \left({t}\right)+\left({t}^{\mathrm{2}} −\nu^{\mathrm{2}} \right){y}\left({t}\right)=\mathrm{0} \\ $$$$\mathrm{we}\:\mathrm{Already}\:\mathrm{Know} \\ $$$$\mathrm{Solution}\:{y}\left({t}\right)={C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {J}_{−\nu} \left({t}\right) \\ $$$$\mathrm{But}\:{J}_{−\nu} \left({t}\right)\:\mathrm{can}'\mathrm{t}\:\mathrm{Satisfy}\:\mathrm{as}\:\mathrm{Solution} \\ $$$$\mathrm{Cus}\:{J}_{\nu} \left({t}\right)\:\mathrm{and}\:{J}_{−\nu} \left({t}\right)\:\mathrm{are}\:\mathrm{Not}\:\mathrm{Linear}\:\mathrm{independent}. \\ $$$$\mathrm{Wronskian}\:\mathcal{W}\in\mathrm{mat}\left({m},{m}\right) \\ $$$$\mathrm{det}\:\mathcal{W}=\mathrm{0} \\ $$$$\mathrm{thus}\:\mathrm{Solution}\:{y}\left({t}\right)={C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right) \\ $$$$\mathrm{i}\:\mathrm{already}\:\mathrm{undertand}\:\mathrm{above}\:\mathrm{indentity} \\ $$$$\mathrm{i}\:\mathrm{wrote} \\ $$$$\mathrm{my}\:\mathrm{question}\:\mathrm{is}\:\mathrm{prove}\:\mathrm{Abel}'\mathrm{s}\:\mathrm{identity} \\ $$$$\:\mathcal{W}\left({J}_{\nu} \left({t}\right),{Y}_{\nu} \left({t}\right)\right)=\frac{\mathrm{2}}{\pi{t}} \\ $$$$\boldsymbol{\mathrm{Pls}}\:\boldsymbol{\mathrm{Help}}\:\::\left(\right. \\ $$

Question Number 212595 Answers: 1 Comments: 0

lim_(n→∞) ((1^2 /(n^2 +1))+(2^2 /(n^2 +2))+(3^2 /(n^2 +3))+…+(n^2 /(n^2 +n))−(n/3))=?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}^{\mathrm{2}} }{{n}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{2}^{\mathrm{2}} }{{n}^{\mathrm{2}} +\mathrm{2}}+\frac{\mathrm{3}^{\mathrm{2}} }{{n}^{\mathrm{2}} +\mathrm{3}}+\ldots+\frac{{n}^{\mathrm{2}} }{{n}^{\mathrm{2}} +{n}}−\frac{{n}}{\mathrm{3}}\right)=? \\ $$

Question Number 212593 Answers: 1 Comments: 1

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