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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 )

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

Question Number 62242 Answers: 2 Comments: 1

Find out x,y such that lcm(x,y)=180 ∧ gcd(x,y)=45

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{180}\:\wedge\:\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{45} \\ $$

Question Number 62234 Answers: 1 Comments: 0

Question Number 62232 Answers: 0 Comments: 4

Question Number 62228 Answers: 0 Comments: 2

{ (((√(a+x))+(√(a−y))=2a)),(((√(a−x))+(√(a+y))=2a)) :} a∈R.

$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{y}}}=\mathrm{2}\boldsymbol{\mathrm{a}}}\\{\sqrt{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{y}}}=\mathrm{2}\boldsymbol{\mathrm{a}}}\end{cases}\:\:\:\:\:\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}. \\ $$

Question Number 62227 Answers: 0 Comments: 3

1.∫(√(1+x+x^2 +x^3 ))dx=? 2.∫ ((√(1−tgx))/(sinx)) dx=? 3.∫ e^x .ln(1+(√(1+x^2 )))dx=? 4.∫ ((sinx)/(1+sinx+sin2x)) dx=?

$$\mathrm{1}.\int\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:\:}\boldsymbol{\mathrm{dx}}=? \\ $$$$\mathrm{2}.\int\:\:\:\frac{\sqrt{\mathrm{1}−\boldsymbol{\mathrm{tgx}}}}{\boldsymbol{\mathrm{sinx}}}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$$$\mathrm{3}.\int\:\:\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} .\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)\boldsymbol{\mathrm{dx}}=? \\ $$$$\mathrm{4}.\int\:\:\frac{\boldsymbol{\mathrm{sinx}}}{\mathrm{1}+\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{sin}}\mathrm{2}\boldsymbol{\mathrm{x}}}\:\boldsymbol{\mathrm{dx}}=? \\ $$

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