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Question Number 62343 Answers: 1 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left\{{x}\prod_{{k}=\mathrm{1}} ^{\infty} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\right\}{dx} \\ $$
Question Number 62342 Answers: 1 Comments: 4
$${let}\:{f}\left(\xi\right)\:=\int\:\:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx}\:\:\:{with}\:\:\mathrm{0}<\xi<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{\xi\rightarrow\mathrm{1}} \:\:\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} \theta\:{x}^{\mathrm{2}} }}\:{dx}\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$
Question Number 62341 Answers: 2 Comments: 3
$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{root}\:\mathrm{does}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{x}^{\mathrm{8}} \:−\:\mathrm{x}^{\mathrm{7}} \:+\:\mathrm{2x}^{\mathrm{6}} \:−\:\mathrm{2x}^{\mathrm{5}} \:+\:\mathrm{3x}^{\mathrm{4}} \:−\:\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{4x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\frac{\mathrm{5}}{\mathrm{2}}\:\:=\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{has} \\ $$
Question Number 62340 Answers: 0 Comments: 3
Question Number 62338 Answers: 0 Comments: 0
Question Number 62335 Answers: 0 Comments: 2
$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{xt}} {cos}\left({yt}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:{g}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{xt}} {sin}\left({yt}\right)}{\sqrt{{t}}}\:{dt} \\ $$$${with}\:{x}>\mathrm{0}\:\:{and}\:{y}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−\mathrm{2}{t}} \:{cos}\left({t}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{t}} {cos}\left(\mathrm{2}{t}\right)}{\sqrt{{t}}}\:{dt} \\ $$
Question Number 62334 Answers: 2 Comments: 0
$${if}\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{2}\alpha\beta\:{evaluate}\left(\alpha−\beta\right) \\ $$
Question Number 62332 Answers: 1 Comments: 0
Question Number 62330 Answers: 1 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt}\:\:\:{with}\:\:\:\mathrm{0}<{a}<\mathrm{1} \\ $$
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
$${M}_{{TP}} ={Q}\frac{{D}}{{Z}}\:×\:{f}_{} \\ $$
Question Number 62289 Answers: 3 Comments: 0
Question Number 62281 Answers: 1 Comments: 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
$$\begin{cases}{\frac{\sqrt{\boldsymbol{\mathrm{x}}}}{\boldsymbol{\mathrm{a}}}+\frac{\sqrt{\boldsymbol{\mathrm{y}}}}{\boldsymbol{\mathrm{b}}}=\mathrm{1}}\\{\frac{\sqrt{\boldsymbol{\mathrm{a}}}}{\boldsymbol{\mathrm{x}}}+\frac{\sqrt{\boldsymbol{\mathrm{b}}}}{\boldsymbol{\mathrm{y}}}=\mathrm{1}}\end{cases}\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$
Question Number 62275 Answers: 1 Comments: 0
$$\begin{cases}{\boldsymbol{\mathrm{a}}\sqrt{\boldsymbol{\mathrm{x}}}+\boldsymbol{\mathrm{b}}\sqrt{\boldsymbol{\mathrm{y}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\\{\boldsymbol{\mathrm{x}}\sqrt{\boldsymbol{\mathrm{a}}}+\boldsymbol{\mathrm{y}}\sqrt{\boldsymbol{\mathrm{b}}}=\mathrm{2}\sqrt{\boldsymbol{\mathrm{ab}}}}\end{cases}\:\:\:\:\:\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\mathrm{R}^{+} \\ $$
Question Number 62274 Answers: 1 Comments: 4
$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{x}^{\mathrm{2}} } \:{dx} \\ $$
Question Number 62273 Answers: 0 Comments: 4
$${Mr}.\:{Rasheed}.{Sindhi}\: \\ $$$${I}\:{sense}\:{you}'{re}\:{much}\:{engaged}\:{in}\:{making}\: \\ $$$${olympiad}\:{contents}\:{these}\:{days}\:,\:{I}\:{wish}\: \\ $$$${that}\:{you}\:{join}\:{my}\:{workspace}\:{concerning}\:{that}\:{same}. \\ $$
Question Number 62266 Answers: 1 Comments: 1
$$\int\frac{\mathrm{2}{sin}\left({x}\right)+\mathrm{3}{cos}\left({x}\right)}{\mathrm{3}{sin}\left({x}\right)+\mathrm{4}{cos}\left({x}\right)}{dx} \\ $$
Question Number 62265 Answers: 1 Comments: 1
Question Number 62263 Answers: 1 Comments: 0
Question Number 62262 Answers: 1 Comments: 1
$${find}\:{the}\:{value}\:{of}\: \\ $$$${I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} {sint}}{\sqrt{{t}}}{dt}\:\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} {cos}\left({t}\right)}{\sqrt{{t}}}{dt}\:\:,{study}\:{first}\:{the}\:{convergence}. \\ $$
Question Number 62252 Answers: 0 Comments: 1
$$\int\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)/\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right) \\ $$$$\mathrm{limit}\:=\left\{\:\mathrm{0}>\mathrm{2}\right\} \\ $$
Question Number 62251 Answers: 0 Comments: 1
$$\int\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{1}/\mathrm{2}} \mathrm{dx} \\ $$$$\mathrm{trig}\:\mathrm{substitution}\:\mathrm{only} \\ $$
Question Number 62244 Answers: 1 Comments: 3
$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y},\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\mathrm{gcd}\left(\mathrm{x}^{\mathrm{3}} ,\mathrm{y}^{\mathrm{2}} \right)=\mathrm{gcd}\left(\mathrm{x}^{\mathrm{2}} ,\mathrm{y}^{\mathrm{3}} \right) \\ $$
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