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Question Number 161133 Answers: 1 Comments: 0
$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{n}}\right)}=? \\ $$
Question Number 161123 Answers: 0 Comments: 0
$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}\left(\mathrm{H}_{\mathrm{2}\boldsymbol{\mathrm{n}}-\mathrm{1}} \:-\:\mathrm{2}\:\mathrm{H}_{\boldsymbol{\mathrm{n}}-\mathrm{1}} \right)} \\ $$
Question Number 161130 Answers: 1 Comments: 0
$$\:{Given}\:{P}\left({x}\right)\:{is}\:{polynomial}\:{such}\:{that} \\ $$$$\:{P}\left(\mathrm{3}{x}\right)=\:{P}\:'\left({x}\right).{P}\:''\left({x}\right)\:.\:{Find}\:{the}\:{tangent} \\ $$$$\:{of}\:{curve}\:{y}\:=\:{P}\left({x}\right)\:{parallel}\:{to}\:{the}\:{line} \\ $$$$\:{y}=\:\mathrm{4}{x}β\mathrm{2}.\: \\ $$
Question Number 161126 Answers: 1 Comments: 2
Question Number 161111 Answers: 1 Comments: 1
$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} +\mathrm{2cos}\:{x}β\mathrm{2}}{{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:{a}=? \\ $$
Question Number 161105 Answers: 1 Comments: 1
Question Number 161102 Answers: 1 Comments: 1
Question Number 161101 Answers: 1 Comments: 0
$${solve}: \\ $$$$\:\:\:\int\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} β\mathrm{7}{x}β\mathrm{3}}{dx} \\ $$
Question Number 161100 Answers: 0 Comments: 0
$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}β\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(β\pi\right)=? \\ $$
Question Number 161096 Answers: 0 Comments: 0
$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{different}\:\mathrm{in}\:\mathrm{pairs}\:\mathrm{and}\:\:\mathrm{n};\mathrm{k}\in\mathbb{N}^{\ast} \\ $$$$\frac{\mathrm{log}\:\mathrm{x}^{\boldsymbol{\mathrm{n}}} }{\mathrm{b}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{c}^{\boldsymbol{\mathrm{k}}} }\:=\:\frac{\mathrm{log}\:\mathrm{y}^{\boldsymbol{\mathrm{n}}} }{\mathrm{c}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{a}^{\boldsymbol{\mathrm{k}}} }\:=\:\frac{\mathrm{log}\:\mathrm{z}^{\boldsymbol{\mathrm{n}}} }{\mathrm{a}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{b}^{\boldsymbol{\mathrm{k}}} } \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\sqrt{\boldsymbol{\mathrm{xyz}}} \\ $$
Question Number 161091 Answers: 1 Comments: 0
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt{\mathrm{1}\:-\:\mathrm{x}}\:=\:\mathrm{2x}^{\mathrm{2}} \:-\:\mathrm{1}\:-\:\mathrm{2x}\:\sqrt{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} } \\ $$$$ \\ $$
Question Number 161114 Answers: 0 Comments: 0
$$\:\:{Let}\:{f}\left({x}\right)=\:\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:{for}\:β\frac{\pi}{\mathrm{4}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{4}} \\ $$$$\:{then}\:{Df}^{β\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)=\frac{{a}}{{b}\sqrt{{b}}}\:{so}\:\begin{cases}{{a}=?}\\{{b}=?}\end{cases} \\ $$
Question Number 161089 Answers: 3 Comments: 0
$$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:β\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$
Question Number 161084 Answers: 0 Comments: 0
Question Number 161079 Answers: 1 Comments: 3
Question Number 161076 Answers: 1 Comments: 0
$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\:\left(\mathrm{1}+\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:ββββββββββββ \\ $$$$\:\:\:\:\:\:\:\: \\ $$
Question Number 161075 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{n}}{\left(\:{n}^{\:\mathrm{2}} β\frac{\:\mathrm{1}}{\mathrm{4}}\:\right)^{\:\mathrm{3}} }\:=\:? \\ $$$$ \\ $$
Question Number 161071 Answers: 2 Comments: 0
$${For}\:\:{a},{b},{c}\:>\:\mathrm{0}\:. \\ $$$${Find}\:\:\left({x},{y},{z}\right)\:\:{that}\:\:{satisfy}\:\:{this}\:\:{equation}\:\:{system}\: \\ $$$$\:\:\:{ax}\:+\:{by}\:=\:\left({x}β{y}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{by}\:+\:{cz}\:=\:\left({y}β{z}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{cz}\:+\:{ax}\:=\:\left({z}β{x}\right)^{\mathrm{2}} \\ $$$$ \\ $$
Question Number 161068 Answers: 2 Comments: 0
$$\:\:\:\:\:\int\:\frac{\mathrm{2}{x}}{\left(\mathrm{1}β{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{4}} β\mathrm{1}}}\:{dx}\:=? \\ $$
Question Number 161066 Answers: 2 Comments: 0
$$\:{x}_{\mathrm{1}} \:,{x}_{\mathrm{2}} \:{be}\:{the}\:{roots}\:{of}\:{the}\:{equation}\: \\ $$$$\:\:\:\:\:\:{x}^{\mathrm{2}} +{x}+{m}=\mathrm{0}\:\&\:{x}_{\mathrm{1}} ^{\mathrm{5}} +{x}_{\mathrm{2}} ^{\mathrm{5}} \:=\:\mathrm{2021}. \\ $$$$\:{Find}\:{the}\:{sum}\:{of}\:{the}\:{possible}\:{values} \\ $$$$\:\:{of}\:{m}. \\ $$
Question Number 161065 Answers: 1 Comments: 0
$$\:\begin{cases}{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:+\sqrt[{\mathrm{8}}]{{x}β{abc}}\:=\:{a}}\\{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:β\sqrt[{\mathrm{8}}]{{x}β{abc}}\:=\:{b}}\\{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:β\sqrt[{\mathrm{4}}]{{x}β{abc}}\:=\:{c}}\end{cases} \\ $$$$\:{find}\:\sqrt{{x}+{abc}}\:+\sqrt{{x}β{abc}} \\ $$
Question Number 161061 Answers: 1 Comments: 0
Question Number 161060 Answers: 1 Comments: 2
$$\mathrm{Given}\:\mathrm{sin}\left(\mathrm{5x}β\mathrm{38}\right)=\mathrm{cos}\left(\mathrm{2x}+\mathrm{16}\right),\:\mathrm{0}Β°\leqslant\mathrm{x}\leqslant\mathrm{90}Β°, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$
Question Number 161059 Answers: 0 Comments: 0
$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2xy}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right)\mathrm{dxdy} \\ $$
Question Number 161058 Answers: 1 Comments: 0
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\mathrm{x}\left(\mathrm{y}-\mathrm{1}\right)\mathrm{dx}\:+\:\left(\mathrm{x}+\mathrm{1}\right)\mathrm{dy}\:=\:\mathrm{0} \\ $$$$ \\ $$
Question Number 161039 Answers: 0 Comments: 0
$$\mathrm{let}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\left(\mathrm{1}\:+\:\mathrm{x}\right)\:\mathrm{y}^{''} \left(\mathrm{x}\right)\:+\:\left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{y}^{'} \left(\mathrm{x}\right)\:=\:\frac{\mathrm{1}-\mathrm{x}}{\mathrm{1}+\mathrm{x}}\:\mathrm{y}\left(\mathrm{x}\right) \\ $$$$\mathrm{y}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:,\:\mathrm{y}^{'} \left(\mathrm{0}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\left(\mathrm{y}^{''} \left(\mathrm{x}\right)\:+\:\mathrm{y}^{'} \left(\mathrm{x}\right)\:+\:\mathrm{y}\left(\mathrm{x}\right)\right)\:\mathrm{e}^{-\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$
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