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Question Number 98268    Answers: 0   Comments: 7

let p(x) be a polynomial function of (n−1)^(th) degree and p(k)=k for k=1,2,3,...,n find p(0) and p(n+1). example: n=10

$${let}\:{p}\left({x}\right)\:{be}\:{a}\:{polynomial}\:{function}\:{of} \\ $$$$\left({n}−\mathrm{1}\right)^{{th}} \:{degree}\:{and} \\ $$$${p}\left({k}\right)={k}\:{for}\:{k}=\mathrm{1},\mathrm{2},\mathrm{3},...,{n} \\ $$$${find}\:{p}\left(\mathrm{0}\right)\:{and}\:{p}\left({n}+\mathrm{1}\right). \\ $$$${example}:\:{n}=\mathrm{10} \\ $$

Question Number 98267    Answers: 2   Comments: 0

∀ a,b>0 , a^2 +b^2 =1 prove that ((1/a)+(1/b))((b/(a^2 +1))+(a/(b^2 +1)))≥(8/3)

$$\forall\:{a},{b}>\mathrm{0}\:,\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{1} \\ $$$${prove}\:{that} \\ $$$$\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\left(\frac{{b}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{a}}{{b}^{\mathrm{2}} +\mathrm{1}}\right)\geqslant\frac{\mathrm{8}}{\mathrm{3}} \\ $$

Question Number 98259    Answers: 2   Comments: 4

Question Number 98256    Answers: 2   Comments: 0

∫_0 ^∞ ((log(x))/((√x)(x+1)^2 ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{log}\left({x}\right)}{\sqrt{{x}}\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 98255    Answers: 1   Comments: 1

solve the following initial values DEs 20y′′ + 4y′ +y = 0 ; y(0) = 3.2 and y′(0) = 0

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{initial}\:\mathrm{values} \\ $$$$\mathrm{DEs}\:\mathrm{20y}''\:+\:\mathrm{4y}'\:+\mathrm{y}\:=\:\mathrm{0} \\ $$$$;\:\mathrm{y}\left(\mathrm{0}\right)\:=\:\mathrm{3}.\mathrm{2}\:\mathrm{and}\:\mathrm{y}'\left(\mathrm{0}\right)\:=\:\mathrm{0}\: \\ $$

Question Number 98250    Answers: 0   Comments: 1

Let A= (((2 2)),((1 3)) ) . Find a non singular matrix P such that P^(−1) AP is a diagonal matrix.

$$\mathrm{Let}\:\mathrm{A}=\begin{pmatrix}{\mathrm{2}\:\:\:\:\mathrm{2}}\\{\mathrm{1}\:\:\:\:\mathrm{3}}\end{pmatrix}\:.\:\mathrm{Find}\:\mathrm{a}\:\mathrm{non}\:\mathrm{singular}\:\mathrm{matrix} \\ $$$$\mathrm{P}\:\mathrm{such}\:\mathrm{that}\:\mathrm{P}^{−\mathrm{1}} \mathrm{AP}\:\mathrm{is}\:\mathrm{a}\:\mathrm{diagonal}\:\mathrm{matrix}. \\ $$

Question Number 98249    Answers: 0   Comments: 0

explicit A(θ) =∫_1 ^(+∞) ((ln(lnx))/(x^2 +2xcosθ +1))dx with −π<θ<π

$$\mathrm{explicit}\:\mathrm{A}\left(\theta\right)\:=\int_{\mathrm{1}} ^{+\infty} \:\frac{\mathrm{ln}\left(\mathrm{lnx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{2xcos}\theta\:+\mathrm{1}}\mathrm{dx}\:\:\:\mathrm{with}\:−\pi<\theta<\pi \\ $$

Question Number 98248    Answers: 0   Comments: 0

find the value of ∫_1 ^(+∞) ((ln(lnx))/(x^2 +1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\mathrm{ln}\left(\mathrm{lnx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 98246    Answers: 1   Comments: 0

∫(x/(sin^2 (x−3)))dx

$$\int\frac{{x}}{{sin}^{\mathrm{2}} \left({x}−\mathrm{3}\right)}{dx} \\ $$

Question Number 98245    Answers: 0   Comments: 0

lim_(k→0) ∫_0 ^k (1/(√(cos(x)−cos(k))))dx=?

$$\underset{{k}\rightarrow\mathrm{0}} {{lim}}\int_{\mathrm{0}} ^{{k}} \frac{\mathrm{1}}{\sqrt{{cos}\left({x}\right)−{cos}\left({k}\right)}}{dx}=? \\ $$

Question Number 98244    Answers: 1   Comments: 0

∫_0 ^(2π) xe^(cosx) cos(sinx)dx= 2π^2

$$\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:{xe}^{{cosx}} {cos}\left({sinx}\right){dx}=\:\mathrm{2}\pi^{\mathrm{2}} \\ $$

Question Number 98215    Answers: 3   Comments: 2

Find the nth term of the sequence {a_n } such that ((a_1 + a_2 + ... + a_n )/n) = n + (1/n) (n = 1, 2, 3, ...)

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{nth}}\:\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\:\:\:\:\frac{\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:+\:\:\boldsymbol{\mathrm{a}}_{\mathrm{2}} \:+\:\:...\:\:+\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}}\:\:\:=\:\:\boldsymbol{\mathrm{n}}\:\:+\:\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}}\:\:\left(\boldsymbol{\mathrm{n}}\:\:=\:\:\mathrm{1},\:\:\mathrm{2},\:\:\mathrm{3},\:\:...\right) \\ $$

Question Number 98214    Answers: 2   Comments: 2

Question Number 98208    Answers: 0   Comments: 2

suppose a force given as F_1 = 24 N and F_2 = 50 N act through points AB and AC where OA = 2i +3j , OB = 5i + 6j and OC = 7i + 8j (a) find in vector notation F_1 and F_2 then find thier resultant.

$$\mathrm{suppose}\:\mathrm{a}\:\mathrm{force}\:\mathrm{given}\:\mathrm{as}\:{F}_{\mathrm{1}} \:=\:\mathrm{24}\:{N}\:\mathrm{and}\:{F}_{\mathrm{2}} \:=\:\mathrm{50}\:{N}\:\mathrm{act}\:\mathrm{through}\: \\ $$$$\mathrm{points}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{where}\:\:{OA}\:=\:\mathrm{2}{i}\:+\mathrm{3}{j}\:,\:{OB}\:=\:\mathrm{5}{i}\:+\:\mathrm{6}{j}\:\:\mathrm{and}\: \\ $$$${OC}\:=\:\mathrm{7}{i}\:+\:\mathrm{8}{j} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{find}\:\mathrm{in}\:\mathrm{vector}\:\mathrm{notation}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{thier}\:\mathrm{resultant}. \\ $$

Question Number 98213    Answers: 2   Comments: 1

Question Number 98205    Answers: 1   Comments: 0

Find the nth term of the sequence {a_n } such that a_1 = 1, a_(n + 1) = (1/2)a_n + ((n^2 − 2n − 1)/(n^2 (n + 1)^2 )) (n = 1, 2, 3, ...)

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{nth}}\:\:\boldsymbol{\mathrm{term}}\:\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:\:=\:\:\mathrm{1},\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\frac{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:−\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\:−\:\:\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\:\:\:\left(\boldsymbol{\mathrm{n}}\:\:=\:\:\mathrm{1},\:\:\mathrm{2},\:\:\mathrm{3},\:\:...\right) \\ $$

Question Number 98201    Answers: 1   Comments: 0

Question Number 98286    Answers: 0   Comments: 0

justify: lim_(x→+∞) Σ_(k≥1) (((k−1)! sin(x−(π/2)k))/x^k )=(π/2)

$${justify}: \\ $$$$\underset{{x}\rightarrow+\infty} {{lim}}\:\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{\left({k}−\mathrm{1}\right)!\:{sin}\left({x}−\frac{\pi}{\mathrm{2}}{k}\right)}{{x}^{{k}} }=\frac{\pi}{\mathrm{2}} \\ $$

Question Number 98192    Answers: 2   Comments: 1

Question Number 98191    Answers: 2   Comments: 0

for what value of :{x∣ 0<x<2𝛑}: 4cosec^2 x−9=cotx

$$\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{what}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\::\left\{\boldsymbol{\mathrm{x}}\mid\:\mathrm{0}<\boldsymbol{\mathrm{x}}<\mathrm{2}\boldsymbol{\pi}\right\}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{4}\boldsymbol{\mathrm{cosec}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}−\mathrm{9}=\boldsymbol{\mathrm{cotx}} \\ $$

Question Number 98190    Answers: 1   Comments: 0

Question Number 98189    Answers: 1   Comments: 0

let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) calculate lim_(x→1^+ ) (x−1)ξ(x)

$$\mathrm{let}\:\xi\left(\mathrm{x}\right)\:=\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{x}} } \\ $$$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{1}^{+} } \:\:\:\left(\mathrm{x}−\mathrm{1}\right)\xi\left(\mathrm{x}\right) \\ $$

Question Number 98188    Answers: 1   Comments: 0

let f(x) =((arctan(2x))/(x+3)) 1) calculate f^((n)) (x) snd f^((n)) (0) 2) developp f at integr serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{arctan}\left(\mathrm{2x}\right)}{\mathrm{x}+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{snd}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$

Question Number 98187    Answers: 3   Comments: 0

find arctan(x)+arctany at form of arctan

$$\mathrm{find}\:\mathrm{arctan}\left(\mathrm{x}\right)+\mathrm{arctany}\:\:\mathrm{at}\:\mathrm{form}\:\mathrm{of}\:\mathrm{arctan} \\ $$

Question Number 98186    Answers: 0   Comments: 0

solve xy^((3)) +x^2 y^((2)) +x^3 y^((1)) +x^4 y =e^(−2x)

$$\mathrm{solve}\:\mathrm{xy}^{\left(\mathrm{3}\right)} \:+\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\left(\mathrm{2}\right)} \:+\mathrm{x}^{\mathrm{3}} \mathrm{y}^{\left(\mathrm{1}\right)} \:+\mathrm{x}^{\mathrm{4}} \mathrm{y}\:=\mathrm{e}^{−\mathrm{2x}} \\ $$

Question Number 98185    Answers: 1   Comments: 0

developp at fourier serie g(x) =(2/(3+sin^2 x))

$$\mathrm{developp}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\frac{\mathrm{2}}{\mathrm{3}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}} \\ $$

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