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Question Number 62343    Answers: 1   Comments: 1

calculate ∫_0 ^(π/4) {xΠ_(k=1) ^∞ cos((x/2^k ))}dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left\{{x}\prod_{{k}=\mathrm{1}} ^{\infty} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\right\}{dx} \\ $$

Question Number 62342    Answers: 1   Comments: 4

let f(ξ) =∫ (x^2 /(√(1−ξx^2 )))dx with 0<ξ<1 1) determine a explicit form of f(ξ) 2) calculate lim_(ξ→1) f(ξ) 3) calculate ∫_0 ^(1/2) (x^2 /(√(1−sin^2 θ x^2 ))) dx with 0<θ<(π/2)

$${let}\:{f}\left(\xi\right)\:=\int\:\:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx}\:\:\:{with}\:\:\mathrm{0}<\xi<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{\xi\rightarrow\mathrm{1}} \:\:\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} \theta\:{x}^{\mathrm{2}} }}\:{dx}\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 62341    Answers: 2   Comments: 3

How many real root does the equation x^8 − x^7 + 2x^6 − 2x^5 + 3x^4 − 3x^3 + 4x^2 − 4x + (5/2) = 0 has

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{root}\:\mathrm{does}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{x}^{\mathrm{8}} \:−\:\mathrm{x}^{\mathrm{7}} \:+\:\mathrm{2x}^{\mathrm{6}} \:−\:\mathrm{2x}^{\mathrm{5}} \:+\:\mathrm{3x}^{\mathrm{4}} \:−\:\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{4x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\frac{\mathrm{5}}{\mathrm{2}}\:\:=\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{has} \\ $$

Question Number 62340    Answers: 0   Comments: 3

Question Number 62338    Answers: 0   Comments: 0

Question Number 62335    Answers: 0   Comments: 2

1) calculate f(x,y) =∫_0 ^∞ ((e^(−xt) cos(yt))/(√t)) dt and g(x,y) =∫_0 ^∞ ((e^(−xt) sin(yt))/(√t)) dt with x>0 and y>0 2) find the values of ∫_0 ^∞ ((e^(−2t) cos(t))/(√t)) dt and ∫_0 ^∞ ((e^(−t) cos(2t))/(√t)) dt

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{xt}} {cos}\left({yt}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:{g}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{xt}} {sin}\left({yt}\right)}{\sqrt{{t}}}\:{dt} \\ $$$${with}\:{x}>\mathrm{0}\:\:{and}\:{y}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−\mathrm{2}{t}} \:{cos}\left({t}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{t}} {cos}\left(\mathrm{2}{t}\right)}{\sqrt{{t}}}\:{dt} \\ $$

Question Number 62334    Answers: 2   Comments: 0

if α^2 +β^2 = (α+β)^2 −2αβ evaluate(α−β)

$${if}\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{2}\alpha\beta\:{evaluate}\left(\alpha−\beta\right) \\ $$

Question Number 62332    Answers: 1   Comments: 0

Question Number 62330    Answers: 1   Comments: 1

find the value of ∫_0 ^∞ (t^(a−1) /((1+t)^2 ))dt with 0<a<1

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt}\:\:\:{with}\:\:\:\mathrm{0}<{a}<\mathrm{1} \\ $$

Question Number 62322    Answers: 0   Comments: 2

Question Number 62308    Answers: 1   Comments: 2

Question Number 62291    Answers: 1   Comments: 1

Question Number 62288    Answers: 0   Comments: 0

M_(TP) =Q(D/Z) × f_

$${M}_{{TP}} ={Q}\frac{{D}}{{Z}}\:×\:{f}_{} \\ $$

Question Number 62289    Answers: 3   Comments: 0

Question Number 62281    Answers: 1   Comments: 0

{ ((x^3 +y^3 =3xy)),((x^4 +y^4 =4xy)) :} [x,y≠0]

$$\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{y}}^{\mathrm{3}} =\mathrm{3}\boldsymbol{\mathrm{xy}}}\\{\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\boldsymbol{\mathrm{y}}^{\mathrm{4}} =\mathrm{4}\boldsymbol{\mathrm{xy}}}\end{cases}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\neq\mathrm{0}\right] \\ $$

Question Number 62276    Answers: 1   Comments: 0

{ ((((√x)/a)+((√y)/b)=1)),((((√a)/x)+((√b)/y)=1)) :} a,b∈R^+

$$\begin{cases}{\frac{\sqrt{\boldsymbol{\mathrm{x}}}}{\boldsymbol{\mathrm{a}}}+\frac{\sqrt{\boldsymbol{\mathrm{y}}}}{\boldsymbol{\mathrm{b}}}=\mathrm{1}}\\{\frac{\sqrt{\boldsymbol{\mathrm{a}}}}{\boldsymbol{\mathrm{x}}}+\frac{\sqrt{\boldsymbol{\mathrm{b}}}}{\boldsymbol{\mathrm{y}}}=\mathrm{1}}\end{cases}\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$

Question Number 62275    Answers: 1   Comments: 0

{ ((a(√x)+b(√y)=2(√(ab)))),((x(√a)+y(√b)=2(√(ab)))) :} a,b∈R^+

$$\begin{cases}{\boldsymbol{\mathrm{a}}\sqrt{\boldsymbol{\mathrm{x}}}+\boldsymbol{\mathrm{b}}\sqrt{\boldsymbol{\mathrm{y}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\\{\boldsymbol{\mathrm{x}}\sqrt{\boldsymbol{\mathrm{a}}}+\boldsymbol{\mathrm{y}}\sqrt{\boldsymbol{\mathrm{b}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\end{cases}\:\:\:\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$

Question Number 62274    Answers: 1   Comments: 4

∫_0 ^∞ e^(−x^2 ) dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{x}^{\mathrm{2}} } \:{dx} \\ $$

Question Number 62273    Answers: 0   Comments: 4

Mr. Rasheed.Sindhi I sense you′re much engaged in making olympiad contents these days , I wish that you join my workspace concerning that same.

$${Mr}.\:{Rasheed}.{Sindhi}\: \\ $$$${I}\:{sense}\:{you}'{re}\:{much}\:{engaged}\:{in}\:{making}\: \\ $$$${olympiad}\:{contents}\:{these}\:{days}\:,\:{I}\:{wish}\: \\ $$$${that}\:{you}\:{join}\:{my}\:{workspace}\:{concerning}\:{that}\:{same}. \\ $$

Question Number 62266    Answers: 1   Comments: 1

∫((2sin(x)+3cos(x))/(3sin(x)+4cos(x)))dx

$$\int\frac{\mathrm{2}{sin}\left({x}\right)+\mathrm{3}{cos}\left({x}\right)}{\mathrm{3}{sin}\left({x}\right)+\mathrm{4}{cos}\left({x}\right)}{dx} \\ $$

Question Number 62265    Answers: 1   Comments: 1

Question Number 62263    Answers: 1   Comments: 0

Question Number 62262    Answers: 1   Comments: 1

find the value of I =∫_0 ^∞ ((e^(−t) sint)/(√t))dt and J =∫_0 ^∞ ((e^(−t) cos(t))/(√t))dt ,study first the convergence.

$${find}\:{the}\:{value}\:{of}\: \\ $$$${I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} {sint}}{\sqrt{{t}}}{dt}\:\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} {cos}\left({t}\right)}{\sqrt{{t}}}{dt}\:\:,{study}\:{first}\:{the}\:{convergence}. \\ $$

Question Number 62252    Answers: 0   Comments: 1

∫ln(x+1)/(x^2 −x+1) limit ={ 0>2}

$$\int\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)/\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right) \\ $$$$\mathrm{limit}\:=\left\{\:\mathrm{0}>\mathrm{2}\right\} \\ $$

Question Number 62251    Answers: 0   Comments: 1

∫(x^2 −4)^(1/2) dx trig substitution only

$$\int\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{1}/\mathrm{2}} \mathrm{dx} \\ $$$$\mathrm{trig}\:\mathrm{substitution}\:\mathrm{only} \\ $$

Question Number 62244    Answers: 1   Comments: 3

Find out x,y, such that gcd(x^3 ,y^2 )=gcd(x^2 ,y^3 )

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y},\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\mathrm{gcd}\left(\mathrm{x}^{\mathrm{3}} ,\mathrm{y}^{\mathrm{2}} \right)=\mathrm{gcd}\left(\mathrm{x}^{\mathrm{2}} ,\mathrm{y}^{\mathrm{3}} \right) \\ $$

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