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Question Number 35608 Answers: 0 Comments: 0

let r∈[0,1[ and x from R F(x,r) = (1/(2π)) ∫_0 ^(2π) (((1−r^2 )f(t))/(1−2r cos(t−x) +r^2 ))dt with f ∈ C^0 (R) 2π periodic and ∣∣f∣∣=sup_(t∈R) ∣f(t)∣ prove that F(x,r)= (a_0 /2) + Σ_(n=1) ^∞ r^n (a_n cos(nx) +b_n sin(nx)) with a_n = (1/π) ∫_0 ^(2π) f(t) cos(nt)dt and b_n = (1/π) ∫_0 ^(2π) f(t)sin(nt)dt

$${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\:{from}\:{R}\right.\right. \\ $$$${F}\left({x},{r}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right){f}\left({t}\right)}{\mathrm{1}−\mathrm{2}{r}\:{cos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}} }{dt}\:\:{with} \\ $$$${f}\:\:\in\:{C}^{\mathrm{0}} \left({R}\right)\:\:\mathrm{2}\pi\:{periodic}\:\:{and}\:\:\mid\mid{f}\mid\mid={sup}_{{t}\in{R}} \mid{f}\left({t}\right)\mid \\ $$$$\:{prove}\:{that}\:{F}\left({x},{r}\right)=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\:\sum_{{n}=\mathrm{1}} ^{\infty} {r}^{{n}} \left({a}_{{n}} {cos}\left({nx}\right)\:+{b}_{{n}} {sin}\left({nx}\right)\right) \\ $$$${with}\:{a}_{{n}} =\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{f}\left({t}\right)\:{cos}\left({nt}\right){dt}\:{and} \\ $$$${b}_{{n}} =\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{f}\left({t}\right){sin}\left({nt}\right){dt} \\ $$

Question Number 35606 Answers: 0 Comments: 2

Question Number 35605 Answers: 0 Comments: 0

let r∈[0,1[ and θ ∈ R,x∈ R prove that 1) 1+ 2 Σ_(n=1) ^(+∞) r^n cosθ = ((1−r^2 )/(1−2r cosθ +r^2 )) 2)1 =(1/(2π)) ∫_0 ^(2π) (((1−r^2 ))/(1−2rcos(t−x) +r^2 ))dt

$${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:\theta\:\in\:{R},{x}\in\:{R}\:{prove}\:{that}\right.\right. \\ $$$$\left.\mathrm{1}\right)\:\mathrm{1}+\:\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:{r}^{{n}} {cos}\theta\:=\:\frac{\mathrm{1}−{r}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{r}\:{cos}\theta\:+{r}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\mathrm{1}\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{2}{rcos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}} }{dt} \\ $$

Question Number 35603 Answers: 0 Comments: 0

let x ∈ R and {x}=x −[x] prove that ∫_1 ^(+∞) (({x})/x^2 ) dx is convergent and find its value .

$${let}\:{x}\:\in\:{R}\:\:{and}\:\left\{{x}\right\}={x}\:−\left[{x}\right] \\ $$$${prove}\:{that}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\left\{{x}\right\}}{{x}^{\mathrm{2}} }\:{dx}\:{is}\:{convergent}\:{and}\:{find} \\ $$$${its}\:{value}\:. \\ $$

Question Number 35602 Answers: 2 Comments: 0

Q_1 .p(x) = 3x^3 + 4x^2 +5x − k and (x−1) is a factor of p(x) find the value of k and the remaining two factors. Q_2 . Evaluate Σ_(r=1) ^∞ 3^(2−r) .

$$\mathrm{Q}_{\mathrm{1}} .{p}\left(\mathrm{x}\right)\:=\:\mathrm{3x}^{\mathrm{3}} +\:\mathrm{4x}^{\mathrm{2}} +\mathrm{5x}\:−\:\mathrm{k}\:\mathrm{and}\: \\ $$$$\left(\mathrm{x}−\mathrm{1}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{p}\left(\mathrm{x}\right)\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{and}\:\mathrm{the}\:\mathrm{remaining}\: \\ $$$$\mathrm{two}\:\mathrm{factors}. \\ $$$${Q}_{\mathrm{2}} .\:{Evaluate}\:\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{3}^{\mathrm{2}−{r}} . \\ $$

Question Number 35595 Answers: 1 Comments: 0

Find the surface Area of a solid cone of raduis 3cm and slant height 4cm. (take π=3.1)

$${Find}\:{the}\:{surface}\:{Area}\:{of}\:{a}\:{solid} \\ $$$${cone}\:{of}\:{raduis}\:\mathrm{3}{cm}\:{and}\:{slant}\: \\ $$$${height}\:\mathrm{4}{cm}.\:\left({take}\:\pi=\mathrm{3}.\mathrm{1}\right) \\ $$

Question Number 35594 Answers: 1 Comments: 0

Given that 18,24,and k + 14 are three consecutive terms of an arithmetic progression Find a) the common difference b) the value of k c)the first term if the 4^(th) term is 12. d) the sum of the first twelve terms of the progression.

$${Given}\:{that}\:\mathrm{18},\mathrm{24},{and}\:{k}\:+\:\mathrm{14}\:{are}\: \\ $$$${three}\:{consecutive}\:{terms}\:{of}\:{an}\: \\ $$$${arithmetic}\:{progression}\:{Find} \\ $$$$\left.{a}\right)\:{the}\:{common}\:{difference} \\ $$$$\left.{b}\right)\:{the}\:{value}\:{of}\:{k} \\ $$$$\left.{c}\right){the}\:{first}\:{term}\:{if}\:{the}\:\mathrm{4}^{{th}} \:{term}\:{is} \\ $$$$\mathrm{12}. \\ $$$$\left.{d}\right)\:{the}\:{sum}\:{of}\:{the}\:{first}\:{twelve}\:{terms}\:{of} \\ $$$${the}\:{progression}. \\ $$

Question Number 35593 Answers: 0 Comments: 0

let f(a)=∫_0 ^∞ e^(−( t^2 +(a/t^2 ))) dt witha>0 find the value of f(a).

$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left(\:{t}^{\mathrm{2}} \:+\frac{{a}}{{t}^{\mathrm{2}} }\right)} {dt}\:{witha}>\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:{f}\left({a}\right). \\ $$

Question Number 35590 Answers: 0 Comments: 0

find J = ∫_0 ^1 e^(−ax) ln(1+e^(−bx) )dx with a>0 and b>0 .

$${find}\:\:{J}\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−{ax}} {ln}\left(\mathrm{1}+{e}^{−{bx}} \right){dx}\:{with}\:{a}>\mathrm{0}\:{and} \\ $$$${b}>\mathrm{0}\:. \\ $$

Question Number 35589 Answers: 0 Comments: 1

let I = ∫_0 ^∞ e^(−tx) ∣sint∣dt with x>0 find the value of I .

$${let}\:\:\:{I}\:\:=\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{tx}} \:\mid{sint}\mid{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 35588 Answers: 1 Comments: 1

calculate ∫_0 ^(π/3) ((sinx cos(cosx))/(1+2sin(cosx)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{sinx}\:{cos}\left({cosx}\right)}{\mathrm{1}+\mathrm{2}{sin}\left({cosx}\right)}{dx} \\ $$

Question Number 35587 Answers: 0 Comments: 0

let f(t) =∫_0 ^1 (e^(−t(1+x^2 )) /(1+x^2 ))dx with t≥0 find a simple form of f(t) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{e}^{−{t}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$

Question Number 35586 Answers: 0 Comments: 0

find the value of f(α) = ∫_0 ^∞ ((arctan(αx))/(1+x^2 ))dx with α from R .

$${find}\:{the}\:{value}\:{of}\: \\ $$$${f}\left(\alpha\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{from}\:{R}\:. \\ $$

Question Number 35585 Answers: 0 Comments: 0

let f(x)= ∫_0 ^x sin(cost)dt developp f at integr serie

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{sin}\left({cost}\right){dt} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 35584 Answers: 0 Comments: 0

let f(t) = ∫_0 ^∞ ((arctan(e^(−tx^2 ) ))/x^2 ) dx with t>0 1) study the existence of f(t) 2) calculate f^′ (t)

$${let}\:\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{arctan}\left({e}^{−{tx}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:\:{the}\:{existence}\:{of}\:\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$

Question Number 35583 Answers: 0 Comments: 0

find ∫_0 ^(π/6) (√(x−sinx)) dx

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\sqrt{{x}−{sinx}}\:{dx} \\ $$

Question Number 35582 Answers: 0 Comments: 0

let g(x)= (1/(2+sinx)) , 2π periodic odd developp f at fourier serie .

$${let}\:{g}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}+{sinx}}\:\:\:\:,\:\mathrm{2}\pi\:{periodic}\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 35581 Answers: 0 Comments: 0

let f(x) = (3/(1+2cosx)) , 2π periodic even developp f at fourier serie.

$${let}\:{f}\left({x}\right)\:=\:\:\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}}\:\:,\:\mathrm{2}\pi\:{periodic}\:{even} \\ $$$${developp}\:{f}\:\:{at}\:{fourier}\:{serie}. \\ $$$$ \\ $$

Question Number 35580 Answers: 0 Comments: 0

if (1/(1+cosx)) = (a_0 /2) +Σ_(n=1) ^∞ a_n cos(nx) calculate a_0 and a_n

$${if}\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} \:{cos}\left({nx}\right)\:\:{calculate}\:{a}_{\mathrm{0}} \\ $$$${and}\:{a}_{{n}} \\ $$

Question Number 35579 Answers: 0 Comments: 0

(u_n ) is a arithmetic sequence with u_0 =1 and u_5 =11 1) find the value of S_n = Σ_(k=0) ^n (1/u_k ^2 ) and lim_(n→+∞) S_n 2) find the value of W_n = Σ_(k=1) ^n (1/(u_(k−1) . u_(k+1) )) and lim_(n→+∞) W_n .

$$\left({u}_{{n}} \right)\:{is}\:{a}\:{arithmetic}\:{sequence}\:{with}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and} \\ $$$${u}_{\mathrm{5}} =\mathrm{11}\:\:\:\:\: \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\:{S}_{{n}} \:\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{u}_{{k}} ^{\mathrm{2}} } \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:{W}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{u}_{{k}−\mathrm{1}} .\:{u}_{{k}+\mathrm{1}} } \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{W}_{{n}} \:. \\ $$

Question Number 35567 Answers: 0 Comments: 2

Question Number 35562 Answers: 0 Comments: 2

Is there a backwards “⇒”?

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{backwards}\:``\Rightarrow''? \\ $$

Question Number 35553 Answers: 0 Comments: 0

Question Number 35551 Answers: 0 Comments: 1

Question Number 35550 Answers: 0 Comments: 0

Question Number 35549 Answers: 0 Comments: 1

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