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Question Number 33094    Answers: 0   Comments: 1

let f(x)= (1/(1+x+x^2 )) dvelopp f at integr serie.

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:\:{dvelopp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 33090    Answers: 1   Comments: 0

Question Number 33089    Answers: 1   Comments: 5

The LCM and GCF of three numbers is 360 and 6 respectively. if the two numbers are 18 and 60. find the third number.

$$\:\boldsymbol{\mathrm{T}}\mathrm{he}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{GCF}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\:\mathrm{360}\:\boldsymbol{\mathrm{and}}\:\mathrm{6}\:\boldsymbol{\mathrm{respectively}}.\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}} \\ $$$$\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{are}}\:\mathrm{18}\:\boldsymbol{\mathrm{and}}\:\mathrm{60}. \\ $$$$\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{number}}. \\ $$

Question Number 33088    Answers: 0   Comments: 3

Find the value of Σ_(n=1) ^∞ (n^2 /2^(n−1) )

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$

Question Number 33074    Answers: 1   Comments: 1

find interms of n the sum Σ_(k=0) ^n k^2 C_n ^k

$${find}\:{interms}\:{of}\:{n}\:\:{the}\:{sum}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:\:{C}_{{n}} ^{{k}} \\ $$

Question Number 33073    Answers: 2   Comments: 1

lim_(x→∞) x^2 e^(−x)

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} {e}^{−{x}} \\ $$

Question Number 33072    Answers: 1   Comments: 1

find Σ_(k=0) ^n k C_n ^k .

$${find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{{n}} ^{{k}} \:. \\ $$

Question Number 33069    Answers: 0   Comments: 0

by using residus theorem prove that ∫_0 ^∞ (t^(a−1) /(1+t)) dt = (π/(sin(πa))) with 0<a<1 .

$${by}\:\:{using}\:{residus}\:{theorem}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}\:{dt}\:=\:\frac{\pi}{{sin}\left(\pi{a}\right)}\:{with}\:\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$

Question Number 33064    Answers: 1   Comments: 2

Question Number 33153    Answers: 0   Comments: 1

it is given that Σ_(r=1 ) ^(20) [f(r)−10]=200 and Σ_(r=1) ^(20) [f(r)−10]^2 =2800 find the value of Σ_(r=1) ^(20) [f(r)]^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\:\:\underset{{r}=\mathrm{1}\:} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]=\mathrm{200} \\ $$$${and} \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]^{\mathrm{2}} =\mathrm{2800} \\ $$$${find}\:{the}\:{value}\:{of} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)\right]^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 33052    Answers: 0   Comments: 1

Question Number 33051    Answers: 0   Comments: 3

Question Number 33048    Answers: 0   Comments: 7

Let f:N→R be a function sarisfying following conditions: f(1)=1. f(1)+2f(2)+....+nf(n)=n(n+1)f(n). Then find the value of 49f(49) ?

$${Let}\:{f}:{N}\rightarrow{R}\:{be}\:{a}\:{function}\:{sarisfying} \\ $$$${following}\:{conditions}: \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1}. \\ $$$${f}\left(\mathrm{1}\right)+\mathrm{2}{f}\left(\mathrm{2}\right)+....+{nf}\left({n}\right)={n}\left({n}+\mathrm{1}\right){f}\left({n}\right). \\ $$$${Then}\:{find}\:{the}\:{value}\:{of}\:\mathrm{49}{f}\left(\mathrm{49}\right)\:? \\ $$

Question Number 33043    Answers: 1   Comments: 0

equal squares as large as possible are drawn on a rectangular ceiling board measuring 54cm by 78cm,find (a)The size of the squares (b)The total number of squares

$$\:\boldsymbol{\mathrm{equal}}\:\boldsymbol{\mathrm{squares}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{large}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{possible}} \\ $$$$\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{drawn}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{rectangular}}\:\boldsymbol{\mathrm{ceiling}}\:\boldsymbol{\mathrm{board}} \\ $$$$\:\boldsymbol{\mathrm{measuring}}\:\mathrm{54}\boldsymbol{\mathrm{cm}}\:\boldsymbol{\mathrm{by}}\:\mathrm{78}\boldsymbol{\mathrm{cm}},\boldsymbol{\mathrm{find}} \\ $$$$\:\left(\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{size}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{squares}} \\ $$$$\:\left(\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{total}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{squares}} \\ $$

Question Number 33036    Answers: 1   Comments: 0

If range of f(x)= (x+1)(x+2)(x+3)(x+4)+5 x∈ [−6,6] is [a,b] ,a,b∈N, find a+b ?

$${If}\:{range}\:{of}\: \\ $$$${f}\left({x}\right)=\:\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}+\mathrm{4}\right)+\mathrm{5} \\ $$$${x}\in\:\left[−\mathrm{6},\mathrm{6}\right]\:{is}\:\left[{a},{b}\right]\:,{a},{b}\in{N},\:{find}\:{a}+{b}\:? \\ $$

Question Number 33032    Answers: 1   Comments: 5

f:N→R f(1)=2005. and f(1)+f(2)+......+f(n)= n^2 f(n),n>1. Then f(2004)=?

$${f}:{N}\rightarrow{R} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2005}. \\ $$$${and}\: \\ $$$${f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+......+{f}\left({n}\right)=\:{n}^{\mathrm{2}} \:{f}\left({n}\right),{n}>\mathrm{1}. \\ $$$${Then}\:{f}\left(\mathrm{2004}\right)=? \\ $$

Question Number 33028    Answers: 0   Comments: 0

find the value of∫_0 ^∞ (e^(−[t]) /(t+1))dt .

$${find}\:{the}\:{value}\:{of}\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\left[{t}\right]} }{{t}+\mathrm{1}}{dt}\:\:. \\ $$

Question Number 33027    Answers: 1   Comments: 0

calculate ∫_0 ^∞ (x^3 /(1+x^5 ))dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{5}} }{dx}. \\ $$

Question Number 33026    Answers: 1   Comments: 1

calculate ∫_0 ^∞ ((1+x^4 )/(1+x^6 )) dx .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\:{dx}\:. \\ $$

Question Number 33009    Answers: 2   Comments: 1

help ! ! ! ∫ (dx/(csc(x)−1)) = ? [ my way ] ∫( (dx/((1/(sinx)) − 1)) ) =∫((sinx)/(1−sinx)) dx =−∫ ((sinx−1+1)/(sinx−1)) dx =−∫1+(1/(sinx−1)) dx =−(∫1dx+∫((sinx+1)/((sinx−1)(sinx+1))) dx) =−(x+C−∫((sinx+1)/(1−sin^2 x)) dx) =−(x+C−∫ ((sinx)/(cos^2 x)) dx−∫ (1/(cos^2 x)) dx) =−(x+C+∫(cosx)^(−2) dcosx−∫(1/(cos^2 x))dx) =−(x−(cosx)^(−1) +C−∫(1/(cos^2 x))dx) ...and I can′t solve the ∫(1/(cos^2 x))dx oh i just found that is tanx+C

$${help}\:!\:!\:! \\ $$$$\int\:\frac{{dx}}{{csc}\left({x}\right)−\mathrm{1}}\:=\:? \\ $$$$ \\ $$$$\left[\:{my}\:{way}\:\right] \\ $$$$\int\left(\:\frac{{dx}}{\frac{\mathrm{1}}{{sinx}}\:−\:\mathrm{1}}\:\right) \\ $$$$=\int\frac{{sinx}}{\mathrm{1}−{sinx}}\:{dx} \\ $$$$=−\int\:\frac{{sinx}−\mathrm{1}+\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\ $$$$=−\int\mathrm{1}+\frac{\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\ $$$$=−\left(\int\mathrm{1}{dx}+\int\frac{{sinx}+\mathrm{1}}{\left({sinx}−\mathrm{1}\right)\left({sinx}+\mathrm{1}\right)}\:{dx}\right) \\ $$$$=−\left({x}+{C}−\int\frac{{sinx}+\mathrm{1}}{\mathrm{1}−{sin}^{\mathrm{2}} {x}}\:{dx}\right) \\ $$$$=−\left({x}+{C}−\int\:\frac{{sinx}}{{cos}^{\mathrm{2}} {x}}\:{dx}−\int\:\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}\:{dx}\right) \\ $$$$=−\left({x}+{C}+\int\left({cosx}\right)^{−\mathrm{2}} {dcosx}−\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx}\right) \\ $$$$=−\left({x}−\left({cosx}\right)^{−\mathrm{1}} +{C}−\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx}\right) \\ $$$$...{and}\:{I}\:{can}'{t}\:{solve}\:{the}\:\int\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}{dx} \\ $$$$ \\ $$$${oh}\:{i}\:{just}\:{found}\:{that}\:{is}\:{tanx}+{C} \\ $$

Question Number 33005    Answers: 0   Comments: 1

Given that y= ((sin x)/(1 + cos x)) find (dy/dx) Evaluate ∫_1 ^2 (x + 4)dx

$$\:{Given}\:{that}\: \\ $$$$\:\:\:{y}=\:\frac{{sin}\:{x}}{\mathrm{1}\:+\:{cos}\:{x}}\:{find}\:\frac{{dy}}{{dx}} \\ $$$${Evaluate}\: \\ $$$$\:\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \left({x}\:+\:\mathrm{4}\right){dx} \\ $$

Question Number 32999    Answers: 1   Comments: 1

Question Number 32998    Answers: 0   Comments: 1

calculate Σ_(n=1) ^∞ ((4n+1)/(n^2 (3n+1)^2 )) .

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{4}{n}+\mathrm{1}}{{n}^{\mathrm{2}} \left(\mathrm{3}{n}+\mathrm{1}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 32997    Answers: 0   Comments: 1

calculate Σ_(n=0) ^∞ ((2n+3)/((n+1)^2 (n+2)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{2}{n}+\mathrm{3}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

Question Number 32996    Answers: 0   Comments: 1

find the sequence (v_n ) wich verify v_(n+2) =(√(v_n .v_(n+1) )) .

$${find}\:{the}\:{sequence}\:\left({v}_{{n}} \right)\:{wich}\:{verify}\:\:{v}_{{n}+\mathrm{2}} \:=\sqrt{{v}_{{n}} \:.{v}_{{n}+\mathrm{1}} }\:. \\ $$

Question Number 32995    Answers: 0   Comments: 1

let u_0 =a , u_1 =b and u_(n+2) =(1/2)(u_n +u_(n+1) ) 1) find u_n interms of n 2) find lim_(n→∞) u_n if a=0

$${let}\:{u}_{\mathrm{0}} ={a}\:,\:{u}_{\mathrm{1}} ={b}\:{and}\:{u}_{{n}+\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} \:+{u}_{{n}+\mathrm{1}} \right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{u}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} \:\:{if}\:{a}=\mathrm{0} \\ $$$$ \\ $$

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