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let give x∈]0,2π[ and a ∈R,b∈ R prove that ((π−x)/2) = arctan(((sinx)/(1−cosx))) 2) prove that ∣arctan(a)−arctan(b)∣≤∣a−b∣ 3)letθ ∈]0,(π/2)[ , x ∈[θ,2π−θ] , r∈[0,1[ prove that ∣ϕ(x,r) −((π−x)/2)∣≤ ((1−r)/((1−cosθ)^2 ))

$$\left.{let}\:{give}\:{x}\in\right]\mathrm{0},\mathrm{2}\pi\left[\:\:{and}\:{a}\:\in{R},{b}\in\:{R}\right. \\ $$$${prove}\:{that}\:\:\frac{\pi−{x}}{\mathrm{2}}\:=\:{arctan}\left(\frac{{sinx}}{\mathrm{1}−{cosx}}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\mid{arctan}\left({a}\right)−{arctan}\left({b}\right)\mid\leqslant\mid{a}−{b}\mid \\ $$$$\left.\mathrm{3}\left.\right){let}\theta\:\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\:\:,\:{x}\:\in\left[\theta,\mathrm{2}\pi−\theta\right]\:,\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{prove}\:{that}\right.\right.\right. \\ $$$$\mid\varphi\left({x},{r}\right)\:−\frac{\pi−{x}}{\mathrm{2}}\mid\leqslant\:\:\frac{\mathrm{1}−{r}}{\left(\mathrm{1}−{cos}\theta\right)^{\mathrm{2}} } \\ $$

Question Number 35609 Answers: 0 Comments: 0

let r ∈[0,1[ and x∈ R and ϕ(x,r) = arctan( ((rsinx)/(1−r cosx))) 1) prove that (∂ϕ/∂x)(x,r) =Σ_(n=1) ^∞ r^n cos(nx) 2)prove that ϕ(x,r) = Σ_(n=1) ^∞ r^n ((sin(nx))/n)

$${let}\:{r}\:\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\in\:{R}\:\:{and}\:\right.\right. \\ $$$$\varphi\left({x},{r}\right)\:=\:{arctan}\left(\:\frac{{rsinx}}{\mathrm{1}−{r}\:{cosx}}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\frac{\partial\varphi}{\partial{x}}\left({x},{r}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:{cos}\left({nx}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\varphi\left({x},{r}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:\:\frac{{sin}\left({nx}\right)}{{n}} \\ $$$$ \\ $$

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