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Question Number 67697 Answers: 1 Comments: 0

⋓si⋒g ChineseRemainderTheorm ∂etermine polynomial p(x) such that p(x)≡8(mod x+1) p(x)≡−24(mod x+3) p(x)≡6(mod x) p(x)≡0(mod x+2)

$$\Cup\mathrm{si}\Cap\mathrm{g}\:\mathrm{ChineseRemainderTheorm} \\ $$$$\partial\mathrm{etermine}\:\mathrm{polynomial}\:\mathrm{p}\left(\mathrm{x}\right)\:\mathrm{such}\:\mathrm{that} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{p}\left(\mathrm{x}\right)\equiv\mathrm{8}\left(\mathrm{mod}\:\mathrm{x}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{p}\left(\mathrm{x}\right)\equiv−\mathrm{24}\left(\mathrm{mod}\:\mathrm{x}+\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{p}\left(\mathrm{x}\right)\equiv\mathrm{6}\left(\mathrm{mod}\:\mathrm{x}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{p}\left(\mathrm{x}\right)\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{x}+\mathrm{2}\right) \\ $$$$ \\ $$

Question Number 67696 Answers: 0 Comments: 0

∫_0 ^1 (1−x)/(1+x)lnx dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}\right)/\left(\mathrm{1}+{x}\right){lnx}\:{dx} \\ $$

Question Number 67692 Answers: 1 Comments: 0

Question Number 67688 Answers: 0 Comments: 1

A relation R defined by _((x,y)) R_((u,v)) ⇔ v^2 −y^2 = u^2 −x^2 show that R is an equivalent Relation.

$${A}\:{relation}\:\mathbb{R}\:{defined}\:{by}\:\:\:_{\left({x},{y}\right)} {R}_{\left({u},{v}\right)} \:\Leftrightarrow\:\:{v}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:{u}^{\mathrm{2}} −{x}^{\mathrm{2}} \\ $$$${show}\:{that}\:{R}\:{is}\:{an}\:{equivalent}\:{Relation}. \\ $$

Question Number 67687 Answers: 0 Comments: 3

find the range of values of ∣((x^2 −9)/3_ )∣= ((9−x^2 )/3)

$${find}\:{the}\:{range}\:{of}\:{values}\:{of}\: \\ $$$$\:\:\mid\frac{{x}^{\mathrm{2}} −\mathrm{9}}{\mathrm{3}_{\:} }\mid=\:\frac{\mathrm{9}−{x}^{\mathrm{2}} }{\mathrm{3}} \\ $$$$ \\ $$

Question Number 67686 Answers: 0 Comments: 2

given that the roots of the equation 4x^2 + 6x + 9 =0 are λ and δ where λ = (1 + α^2 +β^2 ) and δ = α^3 + β^3 find an equation whose roots are (1/(αλ)) and (1/(βδ))

$${given}\:{that}\:{the}\:{roots}\:{of}\:{the}\:{equation}\:\:\mathrm{4}{x}^{\mathrm{2}} \:+\:\mathrm{6}{x}\:+\:\mathrm{9}\:=\mathrm{0}\:{are}\:\:\lambda\:{and}\:\delta\:\:{where}\: \\ $$$$\:\lambda\:=\:\left(\mathrm{1}\:+\:\alpha^{\mathrm{2}} \:+\beta^{\mathrm{2}} \right)\:\:{and}\:\:\delta\:=\:\alpha^{\mathrm{3}} \:+\:\beta^{\mathrm{3}} \\ $$$${find}\:{an}\:{equation}\:{whose}\:{roots}\:{are}\: \\ $$$$\:\:\frac{\mathrm{1}}{\alpha\lambda}\:{and}\:\:\frac{\mathrm{1}}{\beta\delta} \\ $$

Question Number 67684 Answers: 0 Comments: 2

given the function f(x) = { ((x^2 , for 0≤ x< 2)),((ax + 3, for 2≤ x < 4)) :} is periodic of period 4, and is continuous. a) Find the value of a. b) Find the valu of f(6) c) sketch the graph for y =f(x). help me please, for the graph i don′t know wbere to put y=x^2 and y = ax + 3 and where do i put a closed dot and an open dot.

$${given}\:{the}\:{function}\: \\ $$$${f}\left({x}\right)\:=\begin{cases}{{x}^{\mathrm{2}} \:\:,\:{for}\:\:\:\mathrm{0}\leqslant\:{x}<\:\mathrm{2}}\\{{ax}\:+\:\mathrm{3},\:{for}\:\:\mathrm{2}\leqslant\:{x}\:<\:\mathrm{4}}\end{cases} \\ $$$${is}\:{periodic}\:{of}\:{period}\:\:\mathrm{4},\:{and}\:{is}\:{continuous}. \\ $$$$\left.{a}\right)\:{Find}\:\:{the}\:{value}\:{of}\:\:{a}. \\ $$$$\left.{b}\right)\:{Find}\:{the}\:{valu}\:{of}\:\:{f}\left(\mathrm{6}\right) \\ $$$$\left.{c}\right)\:{sketch}\:{the}\:{graph}\:{for}\:{y}\:={f}\left({x}\right). \\ $$$${help}\:{me}\:{please},\:{for}\:{the}\:{graph}\:{i}\:{don}'{t}\:{know}\:{wbere}\:{to}\:{put}\:\:{y}={x}^{\mathrm{2}} \:{and}\:{y}\:=\:{ax}\:+\:\mathrm{3}\:{and} \\ $$$${where}\:{do}\:{i}\:{put}\:{a}\:{closed}\:\:{dot}\:{and}\:{an}\:{open}\:{dot}. \\ $$$$ \\ $$

Question Number 67674 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a))) with a>0 1) determine a explicit form of f(a) 2) calculate g(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a)^2 )) 3)give f^((n)) (a) at form of integral 4)calculate ∫_0 ^∞ (dx/((x^2 +1)(x^2 +3)^2 )) and ∫_0 ^∞ (dx/((x^2 +1)^3 ))

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +{a}\right)}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({a}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 67673 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)(x^2 +x+1)))

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)} \\ $$

Question Number 67672 Answers: 0 Comments: 2

decompose the folowing fraction at R(x) 1)F(x)=(x^3 /(1−x^6 )) 2) G(x) =((x^2 +1)/(x^3 (x^2 +x+1)^2 ))

$${decompose}\:{the}\:{folowing}\:\:{fraction}\:{at}\:{R}\left({x}\right) \\ $$$$\left.\mathrm{1}\right){F}\left({x}\right)=\frac{{x}^{\mathrm{3}} }{\mathrm{1}−{x}^{\mathrm{6}} } \\ $$$$\left.\mathrm{2}\right)\:{G}\left({x}\right)\:=\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 67664 Answers: 0 Comments: 2

show that ∃ n ∈ N^(+ ) : sin^n x + cos^n x = 1 and cosh^n x − sinh^n x = 1. Hint: use Induction method.

$${show}\:{that}\:\:\exists\:{n}\:\in\:{N}^{+\:} \::\:\:{sin}^{{n}} {x}\:+\:{cos}^{{n}} {x}\:=\:\mathrm{1}\:{and}\:\:{cosh}^{{n}} {x}\:−\:{sinh}^{{n}} {x}\:=\:\mathrm{1}. \\ $$$$ \\ $$$${Hint}:\:{use}\:{Induction}\:{method}. \\ $$$$ \\ $$

Question Number 67662 Answers: 0 Comments: 2

please explain the fact that ∫(1/x)dx = ln x + k

$${please}\:{explain}\:{the}\:{fact}\:{that}\: \\ $$$$\int\frac{\mathrm{1}}{{x}}{dx}\:=\:{ln}\:{x}\:+\:{k} \\ $$

Question Number 67660 Answers: 1 Comments: 3

Question Number 67659 Answers: 0 Comments: 3

Help me obtain the value of e from (1 + (1/n))^n how do i go about it.

$${Help}\:{me}\:{obtain}\:\:\:{the}\:{value}\:{of}\:\:{e}\:{from} \\ $$$$\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{n}}\right)^{{n}} \:\:{how}\:{do}\:{i}\:{go}\:{about}\:{it}. \\ $$

Question Number 67655 Answers: 0 Comments: 3

Question Number 67653 Answers: 1 Comments: 0

Question Number 67651 Answers: 0 Comments: 0

1)Let consider S= Σ_(n=0) ^∞ n and T=Σ_(n=0) ^∞ (−1)^(n+1) n We know that ∀ x∈]−1;1] Σ_(n=0) ^∞ (−x)^n =(1/(1+x)) , then after derivating (1/((1+x)^2 ))=Σ_(n=1) ^∞ (−1)^(n+1) nx^(n−1) for x=1 ,we get T=(1/4) Now let ascertain something T=Σ_(n=0) ^∞ (−1)^(2n+2) (2n+1) +Σ_(n=0) ^∞ (−1)^(2n+1) (2n) =Σ_(n=0) ^∞ (2n+1) −2S (•) knowing that S=Σ_(n=0) ^∞ (2n)+Σ_(n=0) ^∞ (2n+1) So Σ_(n=0) ^∞ (2n+1)= S−2S When replacing that value in (•) we get T=(S−2S )−2S If i conclude that T=−3S and finally find S=−(T/3)=−(1/(12)) where will the mistake be? 2)Let consider K=1+((2020)/(2019))+(((2020)/(2019)))^2 +(((2020)/(2019)))^3 +......+... ((2020)/(2019))K=((2020)/(2019))+(((2020)/(2019)))^2 +(((2020)/(2019)))^3 +.... K−1=((2020)/(2019))+(((2020)/(2019)))^2 +(((2020)/(2019)))^3 +... So ((2020)/(2019))K=K−1 Then K=−2019 Where is the error? 3)Let consider n an integer We have 0=n−n =n+(−n)=n+(−n)^(2×(1/2)) =n+[(−n)^2 ]^(1/2) =n+ (√n^2 ) = n+n=2n So << all integer are null : 0 is the only integer>> Where is the error? 4) let consider n an integer different of zero and f(n)=nln(n) we have (df/dn)=ln(n)+1 (•) Likewise f(n)=ln(n^n ) and we know that n^n =n×n×n×......×n (n times) So f(n)=ln(n)+ln(n)+......+ln(n) (n times) Now we have (df/dn)=(1/n)+(1/n)+....+(1/n) (n times) So (df/( dn))=1 (••) Relation (•) and (••) give ln(n)+1=1 then ln(n)=0 ⇒ n=1 << The logarithm of all n≥1 is null : There is no integer big than 1 >> Where is the error? 5) let consider x=0,999999999....... we ascertain that 10x=9,999999999...... then 10x=9+0,999999999.... So 10x=9+x finally x=1 << 0.9999999999999999 ..... is and integer >> Is there any error? 6)let consider a=((26666666666666666)/(66666666666666665)) b=((999999999999999999999995)/(199999999999999999999999)) In the way to cancel , if i just remove one common figure to the numerator and to the denominator And i find a=(2/5) and b=(5/1) Will it be wrong? if no ,explain!

Question Number 67647 Answers: 0 Comments: 0

Question Number 67631 Answers: 0 Comments: 3

Can you please tell me, where does this formula come from? And what means the factorial of a non- integer number? π = ((1/2)!)^2 × 4 I′ve verified the above equation with calculator. Thank you

$$ \\ $$$$ \\ $$$$\:\:\:\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{tell}\:\mathrm{me},\:\mathrm{where}\:\mathrm{does}\:\mathrm{this}\: \\ $$$$\:\:\:\mathrm{formula}\:\mathrm{come}\:\mathrm{from}? \\ $$$$\:\:\:\mathrm{And}\:\mathrm{what}\:\mathrm{means}\:\mathrm{the}\:\mathrm{factorial}\:\mathrm{of}\:\mathrm{a}\:\mathrm{non}- \\ $$$$\:\:\:\mathrm{integer}\:\mathrm{number}? \\ $$$$ \\ $$$$\:\:\:\:\:\:\pi\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}!\right)^{\mathrm{2}} ×\:\mathrm{4} \\ $$$$ \\ $$$$\:\:\:{I}'{ve}\:{verified}\:{the}\:{above}\:{equation}\:{with}\:{calculator}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Thank}\:\mathrm{you} \\ $$$$ \\ $$$$ \\ $$

Question Number 67628 Answers: 0 Comments: 0

∫x^n lnx/n^(x ) dx

$$\int{x}^{{n}} {lnx}/{n}^{{x}\:} \:{dx} \\ $$

Question Number 67623 Answers: 1 Comments: 5