Question and Answers Forum

All Questions   Topic List

Relation and FunctionsQuestion and Answers: Page 11

Question Number 143702    Answers: 2   Comments: 0

n ∈ IN. I_n = ∫_1 ^( e) x^(n+1) lnx dx. 1. prove that (I_n ) is positive and increasing. 2. using a part−by−part integration, calculate I_n .

$${n}\:\in\:\mathrm{IN}. \\ $$$${I}_{{n}} \:=\:\int_{\mathrm{1}} ^{\:\mathrm{e}} {x}^{{n}+\mathrm{1}} {lnx}\:{dx}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\left(\boldsymbol{{I}}_{\boldsymbol{{n}}} \right)\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{increasing}}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{part}}−\boldsymbol{\mathrm{by}}−\boldsymbol{\mathrm{part}}\:\boldsymbol{\mathrm{integration}},\:\boldsymbol{\mathrm{calculate}}\:\boldsymbol{{I}}_{\boldsymbol{{n}}} . \\ $$

Question Number 143606    Answers: 1   Comments: 0

Question Number 143576    Answers: 0   Comments: 0

find L(((arctanx)/x))

$${find}\:{L}\left(\frac{{arctanx}}{{x}}\right) \\ $$

Question Number 143575    Answers: 1   Comments: 0

find L(e^(−(√x)) )

$${find}\:{L}\left({e}^{−\sqrt{{x}}} \right) \\ $$

Question Number 143562    Answers: 0   Comments: 0

Question Number 143546    Answers: 1   Comments: 0

calculate ∫_0 ^∞ ((log^2 x)/((8+x^4 )^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{log}^{\mathrm{2}} \mathrm{x}}{\left(\mathrm{8}+\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 143488    Answers: 1   Comments: 0

let f(x)=(1/(2+sinx)) developp f at fourier serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{2}+\mathrm{sinx}} \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 143383    Answers: 0   Comments: 0

calculate ∫_1 ^3 ((√x)/( (√(x+1))+(√(2x+3))))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\frac{\sqrt{{x}}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{\mathrm{2}{x}+\mathrm{3}}}{dx} \\ $$

Question Number 143382    Answers: 0   Comments: 0

find ∫_0 ^∞ xe^(−x^2 ) log(1+2x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} {xe}^{−{x}^{\mathrm{2}} } {log}\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 143381    Answers: 2   Comments: 0

let f(x)=arctan((√2)x^2 ) 1) calculate f^((n)) (x)and f^((n)) (0) 2)if f(x)=Σa_n x^n find the sequence a_n

$${let}\:{f}\left({x}\right)={arctan}\left(\sqrt{\mathrm{2}}{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right){and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){if}\:{f}\left({x}\right)=\Sigma{a}_{{n}} {x}^{{n}} \:\:{find}\:{the}\: \\ $$$${sequence}\:{a}_{{n}} \\ $$

Question Number 143380    Answers: 1   Comments: 0

developp at fourier serie f(x)=(3/(1+2cosx)) by use of two methods

$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}} \\ $$$${by}\:{use}\:{of}\:{two}\:{methods} \\ $$

Question Number 143262    Answers: 1   Comments: 0

developp at fourier serie f(x)=(1/(cosx +2sinx))

$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}\:+\mathrm{2}{sinx}} \\ $$

Question Number 143261    Answers: 1   Comments: 0

find Y_n =∫_0 ^∞ (dx/((x+1)(x+2)....(x+n))) (n>1 integr)

$${find}\:{Y}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)....\left({x}+{n}\right)} \\ $$$$\left({n}>\mathrm{1}\:{integr}\right) \\ $$

Question Number 143260    Answers: 0   Comments: 1

find ∫ (dx/( (√x)+(√(x+1))+(√(x+2))))

$${find}\:\int\:\frac{{dx}}{\:\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}+\sqrt{{x}+\mathrm{2}}} \\ $$

Question Number 143259    Answers: 1   Comments: 0

solve y^(′′) −y^′ +2=xsin(3x)

$${solve}\:{y}^{''} −{y}^{'} +\mathrm{2}={xsin}\left(\mathrm{3}{x}\right) \\ $$

Question Number 143258    Answers: 1   Comments: 0

calculate lim_(x→1) ∫_x ^x^2 ((sh(xt))/(x+t))dt

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{sh}\left({xt}\right)}{{x}+{t}}{dt} \\ $$

Question Number 143257    Answers: 0   Comments: 0

find ∫∫_([0,1]) e^(−(x^2 +y^2 )) arctan(2(√(x^2 +y^2 )))dxdy

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} {e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {arctan}\left(\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right){dxdy} \\ $$

Question Number 143256    Answers: 0   Comments: 0

calculate ∫_0 ^1 ∫_x ^(2−x) e^(−xy) (√(x+y))dy dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{{x}} ^{\mathrm{2}−{x}} {e}^{−{xy}} \sqrt{{x}+{y}}{dy}\:{dx} \\ $$

Question Number 143255    Answers: 2   Comments: 0

find lim_(x→0) ((sin(1−cosx)+1−cos(sinx))/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{sin}\left(\mathrm{1}−{cosx}\right)+\mathrm{1}−{cos}\left({sinx}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 143254    Answers: 1   Comments: 0

find the value of Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1)(n+2)(n+3)))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)} \\ $$

Question Number 143253    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((arctan(x^3 ))/(1+x^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}^{\mathrm{3}} \right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 143148    Answers: 0   Comments: 0

find v_n =Σ_(k=0) ^n (1/(3k+1)) interms of H_n H_n =Σ_(k=1) ^n (1/k)

$$\mathrm{find}\:\mathrm{v}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{3k}+\mathrm{1}}\:\mathrm{interms}\:\mathrm{of}\:\mathrm{H}_{\mathrm{n}} \\ $$$$\mathrm{H}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{k}} \\ $$

Question Number 143147    Answers: 0   Comments: 1

montrer que lasuite U_n =(H_n /n^2 ) est bornee H_n =Σ_(k=1) ^n (1/n^2 )

$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{lasuite}\:\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{H}_{\mathrm{n}} }{\mathrm{n}^{\mathrm{2}} }\:\mathrm{est}\:\mathrm{bornee} \\ $$$$\mathrm{H}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 142988    Answers: 0   Comments: 0

find the sequence u_n wich verify u_n +u_(n+1) =(2/( (√n))) give a equivalent of u_n (n→∞)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}} +\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\frac{\mathrm{2}}{\:\sqrt{\mathrm{n}}} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{equivalent}\:\mathrm{of}\:\mathrm{u}_{\mathrm{n}} \:\:\left(\mathrm{n}\rightarrow\infty\right) \\ $$

Question Number 142987    Answers: 1   Comments: 0

find the sequence u_n wich verify u_(n+1) =u_n −λu_(n−1) λ real

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} −\lambda\mathrm{u}_{\mathrm{n}−\mathrm{1}} \\ $$$$\lambda\:\mathrm{real} \\ $$

Question Number 142980    Answers: 0   Comments: 0

find U_n =∫_0 ^∞ e^(−nx^2 ) log(2+e^x )dx (n≥1) determine nature of Σ U_n and Σ nU_n

$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } \mathrm{log}\left(\mathrm{2}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx}\:\:\:\left(\mathrm{n}\geqslant\mathrm{1}\right) \\ $$$$\mathrm{determine}\:\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \:\mathrm{and}\:\Sigma\:\mathrm{nU}_{\mathrm{n}} \\ $$

  Pg 6      Pg 7      Pg 8      Pg 9      Pg 10      Pg 11      Pg 12      Pg 13      Pg 14      Pg 15   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com