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Question Number 161100 Answers: 0 Comments: 0
$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}−\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(−\pi\right)=? \\ $$
Question Number 161089 Answers: 3 Comments: 0
$$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:−\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$
Question Number 161076 Answers: 1 Comments: 0
$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\:\left(\mathrm{1}+\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:−−−−−−−−−−−− \\ $$$$\:\:\:\:\:\:\:\: \\ $$
Question Number 161003 Answers: 0 Comments: 0
Question Number 160982 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\:\left(−{ln}\:\left({x}\right)\right)}{\mathrm{1}+{x}}\:{dx}\:\overset{?} {=}\frac{−\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right) \\ $$
Question Number 160979 Answers: 1 Comments: 0
$$\Omega=\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}=???\:\:\: \\ $$$$\boldsymbol{\mathrm{n}}\geqslant\mathrm{1} \\ $$
Question Number 160928 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:{calculate} \\ $$$$ \\ $$$$\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\zeta\:\left(\:\mathrm{1}+\:{n}\:\right)\:−\mathrm{1}}{{n}\:+\:\mathrm{1}}\:\overset{?} {=}\:\mathrm{1}−\:\gamma\: \\ $$$$\:−−−−−−−−−−− \\ $$
Question Number 160902 Answers: 1 Comments: 0
$$\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}}\:\sqrt{\mathrm{cos}\:^{\mathrm{5}} {x}}}\:=? \\ $$
Question Number 160792 Answers: 2 Comments: 0
$$\:\:\:\int\:\frac{\mathrm{sec}\:\mathrm{x}}{\:\sqrt{\mathrm{1}+\mathrm{2sec}\:\mathrm{x}}}\:\sqrt{\frac{\mathrm{cosec}\:\mathrm{x}−\mathrm{cot}\:\mathrm{x}}{\mathrm{cosec}\:\mathrm{x}+\mathrm{cot}\:\mathrm{x}}}\:\mathrm{dx}\:=? \\ $$
Question Number 160767 Answers: 1 Comments: 0
$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}}\:\mathrm{dx}\:=?\: \\ $$
Question Number 160734 Answers: 0 Comments: 1
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:#\:\mathrm{Advanced}\:\:\:\mathrm{Calculus}\:# \\ $$$$\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{ln}^{\:} \:\left(\:\frac{\mathrm{1}}{\mathrm{1}−\:{x}}\:\:\right)}{{x}}\:\right)^{\:\mathrm{3}} {dx}\:\overset{?} {=}\:\mathrm{3}\:\left(\:\zeta\:\left(\mathrm{2}\:\right)\:+\:\zeta\:\left(\mathrm{3}\:\right)\right) \\ $$$$\:\:\:\:\:\:−−−−\:\:{solution}−−−− \\ $$$$\:\:\:\:\:\:\:\:\Phi\:\overset{\mathrm{I}.\mathrm{B}.\mathrm{P}} {=}\:\left[\:\frac{\:\mathrm{1}}{\mathrm{2}{x}^{\:\mathrm{2}} }\:{ln}^{\:\mathrm{3}} \left(\:\mathrm{1}−{x}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} +\frac{\mathrm{3}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \:\left(\mathrm{1}−\:{x}\:\right)}{{x}^{\:\mathrm{2}} \:\left(\mathrm{1}\:−\:{x}\:\right)}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:{lim}_{\:\xi\:\rightarrow\mathrm{1}^{−\:} } \:\frac{{ln}^{\:\mathrm{3}} \left(\:\mathrm{1}−\:\xi\:\right)}{\xi^{\:\mathrm{2}} }\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{3}}{\mathrm{2}}\left[\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \left(\:\mathrm{1}−\:{x}\:\right)}{{x}}{dx}\:=\:\mathrm{2}\:\zeta\:\left(\mathrm{3}\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:+\:\frac{\mathrm{3}}{\mathrm{2}}\left[\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \left(\:\mathrm{1}−{x}\right)}{{x}^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:=\:\mathrm{2}\zeta\:\left(\mathrm{2}\:\right)\right]\: \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{3}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \left(\mathrm{1}−\:{x}\right)}{\mathrm{1}−{x}}\:{dx}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\:{lim}_{\:\xi\:\rightarrow\mathrm{1}^{\:−} } \left\{\frac{{ln}^{\:\mathrm{3}} \left(\:\mathrm{1}−\xi\:\right)}{\xi^{\:\mathrm{2}} }\:\:−{ln}^{\:\mathrm{3}} \left(\mathrm{1}−\:\xi\:\right)\:\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{3}}{\mathrm{2}}\:\left(\mathrm{2}\zeta\:\left(\mathrm{3}\:\right)\right)\:\:+\frac{\mathrm{3}}{\mathrm{2}}\:\left(\:\mathrm{2}\zeta\:\left(\mathrm{2}\:\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{3}\left(\:\:\:\zeta\:\left(\mathrm{3}\:\right)\:+\:\mathrm{3}\zeta\:\left(\mathrm{2}\:\right)\:\:\right)\:\:\:\:\:\:\:\blacksquare\:\:\:{m}.{n}\:\:\: \\ $$$$ \\ $$
Question Number 160733 Answers: 0 Comments: 0
Question Number 160807 Answers: 0 Comments: 0
$$−\mathrm{1}\leqslant{a}_{\mathrm{0}} \leqslant{b}_{\mathrm{0}} \leqslant{c}_{\mathrm{0}} \leqslant\mathrm{1} \\ $$$$\forall{n}\in\mathbb{N}\: \\ $$$${a}_{{n}+\mathrm{1}} =\int_{−\mathrm{1}} ^{\mathrm{1}} {min}\left({x},{b}_{{n}} ,{c}_{{n}} \right){dx} \\ $$$${b}_{{n}+\mathrm{1}} =\int_{−\mathrm{1}} ^{\mathrm{1}} {mil}\left({x},{a}_{{n}} ,{c}_{{n}} \right){dx} \\ $$$${c}_{{n}+\mathrm{1}} =\int_{−\mathrm{1}} ^{\mathrm{1}} {max}\left({x},{b}_{{n}} ,{a}_{{n}} \right){dx} \\ $$$${mil}\left({a},{b},{c}\right)\:{est}\:{le}\:{terme}\:{median}\:{de}\:\left({a},{b},{c}\right) \\ $$$${nature}\:{de}\:\left({a}_{{n}} \right),\left({b}_{{n}} \right),\left({c}_{{n}} \right) \\ $$$$ \\ $$
Question Number 160645 Answers: 2 Comments: 1
$$\:\int\:\mathrm{sin}\:^{\mathrm{8}} \mathrm{x}\:\mathrm{dx}\:=? \\ $$
Question Number 160636 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}\:\:} ^{\:\mathrm{1}} \left(\frac{{tanh}^{\:−\mathrm{1}} \left(\:{x}\:\right)}{{x}}\:\right)^{\:\mathrm{2}} =\:\zeta\:\left(\:\mathrm{2}\:\right)\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\: \\ $$$$ \\ $$$$ \\ $$
Question Number 160597 Answers: 1 Comments: 0
$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{dx}}{\mathrm{2}−\mathrm{cos}\:\mathrm{x}}\:=? \\ $$
Question Number 160594 Answers: 1 Comments: 0
$$\:\:\:\:\int\:\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:=? \\ $$
Question Number 160590 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\:{arcsinh}\left({x}\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\pi^{\mathrm{2}} }{\mathrm{20}} \\ $$
Question Number 160563 Answers: 1 Comments: 0
$$\int\mathrm{x}\left\{\mathrm{x}\right\}\left[\mathrm{x}\right]\mathrm{dx}=? \\ $$
Question Number 160556 Answers: 0 Comments: 0
$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{arctg}}\left(\boldsymbol{\mathrm{x}}\right)}{\mathrm{1}+\boldsymbol{\mathrm{x}}}\centerdot\frac{\boldsymbol{\mathrm{dx}}}{\:\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{x}}}}=? \\ $$
Question Number 160551 Answers: 1 Comments: 2
$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{arctan}\:\mathrm{x}\centerdot\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx}=? \\ $$
Question Number 160547 Answers: 1 Comments: 0
$$\:\:{solve} \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \:\right)\sqrt{\:{x}}}\:{dx}=\:? \\ $$$$−−−−−−−− \\ $$
Question Number 160543 Answers: 0 Comments: 0
$$ \\ $$$$\mathrm{lim}_{\:{x}\rightarrow\:\mathrm{6}} \frac{\:\Gamma\:\left(\:{sin}\left(\:\frac{\pi}{{x}}\right)\right)−\Gamma\:\left(\frac{\mathrm{3}}{{x}}\:\right)}{{sin}\left(\:\pi{x}\:\right)}=\:? \\ $$$$ \\ $$
Question Number 160415 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{t}}^{\boldsymbol{\mathrm{n}}} }{\mathrm{1}+\boldsymbol{\mathrm{t}}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\boldsymbol{\mathrm{dt}}=? \\ $$
Question Number 160362 Answers: 0 Comments: 0
$$\int\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}} }{\:\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}\boldsymbol{\mathrm{dx}}=? \\ $$
Question Number 160358 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{sin}{x}\right)\mathrm{ln}\left(\mathrm{cos}{x}\right){dx} \\ $$
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