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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

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

Question Number 32994    Answers: 0   Comments: 1

find ∫_0 ^∞ ((1−cos(λx))/x^2 ) dx with λ>0 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}−{cos}\left(\lambda{x}\right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32993    Answers: 0   Comments: 0

calculate ∫_0 ^∞ e^(−λx) ((sinx)/(√x)) dx wih λ>0 .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{x}} \:\frac{{sinx}}{\sqrt{{x}}}\:{dx}\:\:{wih}\:\lambda>\mathrm{0}\:. \\ $$

Question Number 32992    Answers: 0   Comments: 0

find lim_(x→0) ((tan(sinx) −sin(tanx))/x^2 ) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{tan}\left({sinx}\right)\:−{sin}\left({tanx}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32991    Answers: 1   Comments: 1

calculate lim_(x→0) ((ln(1+x) −x)/x^2 ) .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)\:−{x}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32990    Answers: 1   Comments: 1

find lim_(x→0^+ ) x e^(−lnx) .

$$\:\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{x}\:{e}^{−{lnx}} \:\:. \\ $$

Question Number 32989    Answers: 0   Comments: 0

let give f(x)=(((√(2+x)) −(√(2−x)) )/x) find f^(−1) (x) and calculate (f^(−1) (x))^′ .

$${let}\:{give}\:{f}\left({x}\right)=\frac{\sqrt{\mathrm{2}+{x}}\:−\sqrt{\mathrm{2}−{x}}\:}{{x}} \\ $$$${find}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{calculate}\:\left({f}^{−\mathrm{1}} \left({x}\right)\right)^{'} \:\:. \\ $$

Question Number 32987    Answers: 1   Comments: 0

If 1=1 2=4 3=10 4=20 5=?

$${If}\:\mathrm{1}=\mathrm{1} \\ $$$$\mathrm{2}=\mathrm{4} \\ $$$$\mathrm{3}=\mathrm{10}\: \\ $$$$\mathrm{4}=\mathrm{20} \\ $$$$\mathrm{5}=? \\ $$

Question Number 32985    Answers: 1   Comments: 0

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