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Question Number 73144 Answers: 1 Comments: 1

calculte ∫ ((x+(√(2+x^2 )))/(x+1−(√(2+x^2 ))))dx

$${calculte}\:\int\:\:\frac{{x}+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{{x}+\mathrm{1}−\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 73137 Answers: 0 Comments: 8

Question Number 73131 Answers: 0 Comments: 2

solve for x,in terms of: a∈R . x+(√x)+(√(x^2 −a))+(√(x−a^2 ))=a^2

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}:\:\:\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}\:. \\ $$$$\:\:\:\boldsymbol{\mathrm{x}}+\sqrt{\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}}+\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} }=\boldsymbol{\mathrm{a}}^{\mathrm{2}} \\ $$

Question Number 73117 Answers: 2 Comments: 1

Question Number 73111 Answers: 1 Comments: 0

Question Number 73113 Answers: 1 Comments: 3

Question Number 73090 Answers: 2 Comments: 1

Question Number 73211 Answers: 0 Comments: 0

y=(c+3)(√x) +((3+d)/x)−((a+4)/x^(a+3) )

$${y}=\left({c}+\mathrm{3}\right)\sqrt{{x}}\:+\frac{\mathrm{3}+{d}}{{x}}−\frac{{a}+\mathrm{4}}{{x}^{{a}+\mathrm{3}} } \\ $$

Question Number 73080 Answers: 1 Comments: 0

y′′=e^y pls solve

$$\mathrm{y}''=\mathrm{e}^{\mathrm{y}} \\ $$$$\mathrm{pls}\:\mathrm{solve} \\ $$

Question Number 73059 Answers: 1 Comments: 3

let P_n (x)=(x+1)^n −(x−1)^n 1) fartorize inside C(x) P_n (x) 2)calculate Π_(k=1) ^p cotan(((kπ)/(2p+1)))

$${let}\:{P}_{{n}} \left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} −\left({x}−\mathrm{1}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{fartorize}\:{inside}\:{C}\left({x}\right)\:{P}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:{cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right) \\ $$

Question Number 73057 Answers: 1 Comments: 0

let P_n =(x+1)^(2n+1) −x^(2n+1) −1 prove that x^2 +x divide P_n

$${let}\:{P}_{{n}} =\left({x}+\mathrm{1}\right)^{\mathrm{2}{n}+\mathrm{1}} −{x}^{\mathrm{2}{n}+\mathrm{1}} −\mathrm{1} \\ $$$${prove}\:{that}\:{x}^{\mathrm{2}} \:+{x}\:{divide}\:\underset{{n}} {{P}} \\ $$

Question Number 73056 Answers: 0 Comments: 1

if (xsina+cosa)^n =q(x^2 +1)+r find r

$${if}\:\left({xsina}+{cosa}\right)^{{n}} ={q}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)+{r}\:{find}\:{r} \\ $$

Question Number 73055 Answers: 0 Comments: 0

decompose inside R(x) the fraction ((x^4 +x+1)/(x(x^2 +1)^3 ))

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction}\:\:\frac{{x}^{\mathrm{4}} \:+{x}+\mathrm{1}}{{x}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 73052 Answers: 1 Comments: 0

let P_n =X^n +X^(n−1) +....+X^2 +X−1 ∈R[X] 1)prove that P_n have one root x_n inside ]0,+∞[ 2)study the sequence x_n

$${let}\:{P}_{{n}} ={X}^{{n}} \:+{X}^{{n}−\mathrm{1}} \:+....+{X}^{\mathrm{2}} \:+{X}−\mathrm{1}\:\in{R}\left[{X}\right] \\ $$$$\left.\mathrm{1}\left.\right){prove}\:{that}\:{P}_{{n}} {have}\:{one}\:{root}\:{x}_{{n}} \:{inside}\:\right]\mathrm{0},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{sequence}\:{x}_{{n}} \\ $$

Question Number 73051 Answers: 0 Comments: 1

factorize inside R[X] 1)X^5 −1 2)X^6 +1

$${factorize}\:{inside}\:{R}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right){X}^{\mathrm{5}} −\mathrm{1}\:\: \\ $$$$\left.\mathrm{2}\right){X}^{\mathrm{6}} \:+\mathrm{1} \\ $$

Question Number 73049 Answers: 0 Comments: 0

simplifyA_n = Σ_(k=0) ^n ((k/n)−α)^2 C_n ^k X^k (1−X)^(n−k)

$${simplifyA}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}−\alpha\right)^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{X}^{{k}} \left(\mathrm{1}−{X}\right)^{{n}−{k}} \\ $$

Question Number 73048 Answers: 0 Comments: 0

solve inside Z^2 x^2 +3xy−2y^2 =122

$${solve}\:{inside}\:{Z}^{\mathrm{2}} \:\:{x}^{\mathrm{2}} \:+\mathrm{3}{xy}−\mathrm{2}{y}^{\mathrm{2}} \:=\mathrm{122} \\ $$

Question Number 73047 Answers: 1 Comments: 0

solve inside Z^3 x^2 +y^2 +z^(2 ) =2xyz

$${solve}\:{inside}\:{Z}^{\mathrm{3}} \:\:\:\:\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}\:} =\mathrm{2}{xyz} \\ $$

Question Number 73046 Answers: 1 Comments: 0

prove that ∀n ∈N Σ_(k=0) ^n k C_(2n) ^(n+k) =nC_(2n−1) ^n

$${prove}\:{that}\:\forall{n}\:\in{N}\:\:\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{\mathrm{2}{n}} ^{{n}+{k}} \:={nC}_{\mathrm{2}{n}−\mathrm{1}} ^{{n}} \\ $$

Question Number 73045 Answers: 0 Comments: 0

calculate Σ_(k=0) ^n (C_n ^k /C_(2n−1) ^k )

$${calculate}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{C}_{{n}} ^{{k}} }{{C}_{\mathrm{2}{n}−\mathrm{1}} ^{{k}} } \\ $$

Question Number 73044 Answers: 0 Comments: 0

prove that Σ_(k=1) ^n H_k =(n+1)H_n −n and Σ_(k=1) ^n H_k ^2 =(n+1)H_n ^2 −(2n+1)H_n +2n

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} −{n} \\ $$$${and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:−\left(\mathrm{2}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$

Question Number 73043 Answers: 1 Comments: 2

prove that H_n =Σ_(k=1) ^n (((−1)^(k+1) )/k)×C_n ^k H_n =Σ_(k=1) ^n (1/k)

$${prove}\:{that}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}}×{C}_{{n}} ^{{k}} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 73042 Answers: 1 Comments: 0

prove that for (n,p)∈N^★^2 Σ_(k=0) ^(p ) k C_n ^(p−k) C_n ^k =n C_(2n−1) ^(p−1) conclude the value of Σ_(k=0) ^n k (C_n ^k )^2

$${prove}\:{that}\:{for}\:\left({n},{p}\right)\in{N}^{\bigstar^{\mathrm{2}} } \:\:\:\sum_{{k}=\mathrm{0}} ^{{p}\:} \:{k}\:{C}_{{n}} ^{{p}−{k}} \:{C}_{{n}} ^{{k}} \:={n}\:{C}_{\mathrm{2}{n}−\mathrm{1}} ^{{p}−\mathrm{1}} \\ $$$${conclude}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} \\ $$

Question Number 73041 Answers: 1 Comments: 1

prove that ∀n∈ N Σ_(k=0) ^(2n) (−1)^k (C_(2n) ^k )^2 =(−1)^n C_(2n) ^n

$${prove}\:{that}\:\forall{n}\in\:{N}\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(−\mathrm{1}\right)^{{k}} \:\left({C}_{\mathrm{2}{n}} ^{{k}} \right)^{\mathrm{2}} \:=\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} \\ $$

Question Number 73040 Answers: 1 Comments: 0

prove that ∀(n,p)∈N^★ ×N 1)Σ_(k=0) ^p (−1)^k C_n ^k =(−1)^p C_(n−1) ^p 2)∀(p,q)∈N^2 Σ_(k=0) ^p C_(p+q) ^k C_(p+q−k) ^(p−k) =2^p C_(p+q) ^p

$${prove}\:{that}\:\:\forall\left({n},{p}\right)\in{N}^{\bigstar} ×{N} \\ $$$$\left.\mathrm{1}\right)\sum_{{k}=\mathrm{0}} ^{{p}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$$$\left.\mathrm{2}\right)\forall\left({p},{q}\right)\in{N}^{\mathrm{2}} \:\:\:\:\sum_{{k}=\mathrm{0}} ^{{p}} \:{C}_{{p}+{q}} ^{{k}} \:{C}_{{p}+{q}−{k}} ^{{p}−{k}} \:\:=\mathrm{2}^{{p}} \:{C}_{{p}+{q}} ^{{p}} \\ $$

Question Number 73039 Answers: 1 Comments: 0

let U_n =(n/2) if n even and U_n =((n−1)/2) if n odd let f(n)=Σ_(k=0) ^n U_k prove that ∀(x,y)∈N^2 f(x+y)−f(x−y)=xy

$${let}\:{U}_{{n}} =\frac{{n}}{\mathrm{2}}\:{if}\:{n}\:{even}\:{and}\:{U}_{{n}} =\frac{{n}−\mathrm{1}}{\mathrm{2}}\:{if}\:{n}\:{odd}\:{let}\:{f}\left({n}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {U}_{{k}} \\ $$$${prove}\:{that}\:\forall\left({x},{y}\right)\in{N}^{\mathrm{2}} \:\:\:\:{f}\left({x}+{y}\right)−{f}\left({x}−{y}\right)={xy} \\ $$

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