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AllQuestion and Answers: Page 1786

Question Number 30593 Answers: 1 Comments: 0

factorize inside C[x] p(x)=(1+i(x/n))^n −(1−i(x/n))^n .

$${factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right)=\left(\mathrm{1}+{i}\frac{{x}}{{n}}\right)^{{n}} \:−\left(\mathrm{1}−{i}\frac{{x}}{{n}}\right)^{{n}} . \\ $$

Question Number 30592 Answers: 1 Comments: 0

let p(x)=x^(2n) −2cosα x^n +1 1) find roots lf p(x) 2)factorize p(x) inside C[x] 3)factorize p(x) inside R[x].

$${let}\:{p}\left({x}\right)={x}^{\mathrm{2}{n}} \:−\mathrm{2}{cos}\alpha\:{x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{lf}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{R}\left[{x}\right]. \\ $$

Question Number 30590 Answers: 1 Comments: 0

decompose sur R[x] x^(2n+1) −1.

$${decompose}\:{sur}\:{R}\left[{x}\right]\:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:−\mathrm{1}. \\ $$

Question Number 30589 Answers: 0 Comments: 0

let U_n = {z∈C / z^n =1} find S= Σ_(z∈U_n ) (z/((x−z)^2 )) .

$${let}\:{U}_{{n}} =\:\left\{{z}\in{C}\:/\:{z}^{{n}} =\mathrm{1}\right\}\:\:{find} \\ $$$${S}=\:\sum_{{z}\in{U}_{{n}} } \:\:\frac{{z}}{\left({x}−{z}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 30588 Answers: 0 Comments: 0

(n_k )_(1≤k≤n) is a family of integrs numbers let put p(x)=Σ_(k=1) ^n x^n_k and q(x)= Σ_(j=0) ^(n−1) x^j if n_k ≡k−1[n] prove that q divide p.

$$\left({n}_{{k}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} \:{is}\:{a}\:{family}\:{of}\:{integrs}\:{numbers}\:{let}\:{put} \\ $$$${p}\left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:{x}^{{n}_{{k}} } \:\:\:{and}\:{q}\left({x}\right)=\:\sum_{{j}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{x}^{{j}} \: \\ $$$${if}\:{n}_{{k}} \equiv{k}−\mathrm{1}\left[{n}\right]\:{prove}\:{that}\:{q}\:{divide}\:{p}. \\ $$

Question Number 30587 Answers: 0 Comments: 0

find Σ_(k=0) ^(n−1) (−1)^k cos^n (((kπ)/n)).

$${find}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\left(−\mathrm{1}\right)^{{k}} \:{cos}^{{n}} \left(\frac{{k}\pi}{{n}}\right). \\ $$

Question Number 30586 Answers: 0 Comments: 0

let p=1+x+x^2 +....+x^(2^(n+1) −1) and q= 1+x^2^n find α= (p/q) .

$${let}\:{p}=\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+....+{x}^{\mathrm{2}^{{n}+\mathrm{1}} −\mathrm{1}} \:{and}\:\:{q}=\:\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \\ $$$${find}\:\alpha=\:\frac{{p}}{{q}}\:. \\ $$

Question Number 30585 Answers: 0 Comments: 0

find F_n (x)= ∫_0 ^∞ (x^n /(e^(x+n) +1))dx .

$${find}\:\:{F}_{{n}} \left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{{n}} }{{e}^{{x}+{n}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 30584 Answers: 0 Comments: 0

find I= ∫_(−∞) ^(+∞) (e^(−x^2 ) /(a^2 +(v−x)^2 ))dx.

$${find}\:\:{I}=\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{e}^{−{x}^{\mathrm{2}} } }{{a}^{\mathrm{2}} \:+\left({v}−{x}\right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30583 Answers: 0 Comments: 0

decompose F =(1/((x^2 −1)^n )) inside C[x].n from N.

$${decompose}\:{F}\:=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{{n}} }\:{inside}\:{C}\left[{x}\right].{n}\:{from}\:{N}. \\ $$

Question Number 30582 Answers: 0 Comments: 0

x_1 , x_2 , x_(3 ) are roots of the polynomial x^3 −x+1 find the polynomial wich have for roots x_1 ^3 ,x_2 ^3 and x_3 ^3 .

$${x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}\:} \:{are}\:{roots}\:{of}\:{the}\:{polynomial}\:{x}^{\mathrm{3}} \:−{x}+\mathrm{1}\:{find} \\ $$$${the}\:{polynomial}\:{wich}\:{have}\:{for}\:{roots}\:{x}_{\mathrm{1}} ^{\mathrm{3}} \:,{x}_{\mathrm{2}} ^{\mathrm{3}} \:{and}\:{x}_{\mathrm{3}} ^{\mathrm{3}} \:\:. \\ $$

Question Number 30581 Answers: 0 Comments: 0

decompose inside C[x] F= (1/((x+iy)^n )) .

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\:\frac{\mathrm{1}}{\left({x}+{iy}\right)^{{n}} }\:. \\ $$

Question Number 30580 Answers: 0 Comments: 1

decompose inside C[x] F= (x^n /(x^m +1)) with m≥n+2 then find ∫_0 ^∞ (x^n /(x^m +1))dx.

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}\:{with}\:{m}\geqslant{n}+\mathrm{2} \\ $$$${then}\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{n}} }{{x}^{{m}} \:+\mathrm{1}}{dx}. \\ $$

Question Number 30579 Answers: 0 Comments: 0

decompose inside C[x] F= ((x^n −1)/(x^(2n) −1)) .

$${decompose}\:{inside}\:{C}\left[{x}\right]\:{F}=\:\frac{{x}^{{n}} −\mathrm{1}}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:. \\ $$

Question Number 30578 Answers: 0 Comments: 0

decompose F(x)= (1/((x^2 +1)^n )) on C[x].with n fromN.

$${decompose}\:{F}\left({x}\right)=\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{n}} }\:{on}\:{C}\left[{x}\right].{with}\:{n}\:{fromN}. \\ $$

Question Number 30576 Answers: 0 Comments: 1

let consider the equation x^3 +px +q find S= Σ_(i≠j) (x_i /x_j ) .

$${let}\:{consider}\:{the}\:{equation}\:{x}^{\mathrm{3}} \:+{px}\:+{q}\: \\ $$$${find}\:{S}=\:\sum_{{i}\neq{j}} \:\frac{{x}_{{i}} }{{x}_{{j}} }\:. \\ $$

Question Number 30575 Answers: 0 Comments: 0

find ∫∫_D (x^2 +y^2 )dxdy with D={(x,y)/ x≤1 and x^2 ≤y≤2 }.

$${find}\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}=\left\{\left({x},{y}\right)/\:{x}\leqslant\mathrm{1}\:{and}\:{x}^{\mathrm{2}} \leqslant{y}\leqslant\mathrm{2}\:\right\}. \\ $$

Question Number 30574 Answers: 0 Comments: 0

find ∫∫_([1,e]^2 ) ln(xy)dxdy.

$${find}\:\int\int_{\left[\mathrm{1},{e}\right]^{\mathrm{2}} } \:\:\:{ln}\left({xy}\right){dxdy}. \\ $$

Question Number 30573 Answers: 0 Comments: 0

find ∫∫_([0,1]×[0,1]) (x^2 /(1+y^2 ))dxdy.

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right]} \:\:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{y}^{\mathrm{2}} }{dxdy}. \\ $$

Question Number 30572 Answers: 0 Comments: 1

find I=∫∫_([3,4]×[1,2]) ((dxdy)/((x+y)^2 )) .

$${find}\:\:{I}=\int\int_{\left[\mathrm{3},\mathrm{4}\right]×\left[\mathrm{1},\mathrm{2}\right]} \:\:\frac{{dxdy}}{\left({x}+{y}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 30571 Answers: 0 Comments: 0

decompose F(x)= (1/((1+x+x^2 )^n )) on C[x].

$${decompose}\:{F}\left({x}\right)=\:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)^{{n}} }\:{on}\:{C}\left[{x}\right]. \\ $$

Question Number 30570 Answers: 0 Comments: 0

find ∫∫_U ((dxdy)/(x^2 +y^2 )) with U= {(x,y)∈R^2 /1≤x^2 +2y^2 ≤4}

$${find}\:\int\int_{{U}} \:\frac{{dxdy}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:{with}\:{U}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{1}\leqslant{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \leqslant\mathrm{4}\right\} \\ $$

Question Number 30569 Answers: 0 Comments: 0

find I= ∫∫_D (√(1−(x^2 /a^2 )−(y^2 /b^2 ))) dxdy with D is the interior of ellipce (x^2 /a^2 ) +(y^2 /b^2 ) =1.

$${find}\:{I}=\:\int\int_{{D}} \:\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }}\:\:{dxdy}\:\:{with}\:{D}\:{is}\:{the}\:{interior} \\ $$$${of}\:{ellipce}\:\:\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\mathrm{1}. \\ $$

Question Number 30568 Answers: 0 Comments: 0

find ∫_(−1) ^1 (dx/((√(1+x^2 )) +(√(1−x^2 )))) .

$${find}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} \:}}\:. \\ $$

Question Number 30567 Answers: 0 Comments: 0

integrate xy^, +(x−1)y +y^2 =0

$${integrate}\:{xy}^{,} \:+\left({x}−\mathrm{1}\right){y}\:+{y}^{\mathrm{2}} =\mathrm{0} \\ $$

Question Number 30566 Answers: 0 Comments: 0

study the convergence of A(α) =∫_0 ^∞ (t^(α−1) /(1+t^2 ))dt and find its value.

$${study}\:{the}\:{convergence}\:{of}\:{A}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{\alpha−\mathrm{1}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and} \\ $$$${find}\:{its}\:{value}. \\ $$

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