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Question Number 151082 Answers: 0 Comments: 1
Question Number 151081 Answers: 3 Comments: 0
$${if}\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${and}\:{f}\left({x}+\mathrm{3}\right)=\mathrm{7}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\:{find} \\ $$$$\left.\mathrm{1}\left.\right)\:{a}+{b}+{c}\:\:\:\:\:\mathrm{2}\right)\:{a}−{b}+{c} \\ $$
Question Number 151080 Answers: 2 Comments: 0
$${if}\:\left({f}\circ{f}\circ{f}\circ{f}\right)\left({x}\right)=\mathrm{16}{x}+\mathrm{15} \\ $$$${find}\:{f}\left({x}\right) \\ $$
Question Number 151079 Answers: 1 Comments: 0
Question Number 151078 Answers: 0 Comments: 0
$$\Omega_{\boldsymbol{\mathrm{k}}} =\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\:\centerdot\underset{\boldsymbol{\mathrm{p}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{p}}\end{pmatrix}}{\begin{pmatrix}{\mathrm{n}+\mathrm{k}}\\{\mathrm{n}+\mathrm{p}}\end{pmatrix}}\:\:;\:\:\mathrm{k}\in\mathbb{N}^{\ast} -\mathrm{fixed} \\ $$$$\mathrm{find}\:\:\Omega=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\Omega_{\boldsymbol{\mathrm{n}}-\mathrm{1}} }\:\centerdot\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\sqrt[{\boldsymbol{\mathrm{i}}^{\mathrm{2}} }]{\boldsymbol{\mathrm{i}}!}\: \\ $$
Question Number 151074 Answers: 1 Comments: 0
$$\mathrm{5}+\mathrm{5}+\mathrm{5} \\ $$
Question Number 151070 Answers: 0 Comments: 4
$$\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{30}\:\:\:\mathrm{and}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{a}}\:+\:\frac{\mathrm{1}}{\mathrm{b}}\:+\:\frac{\mathrm{1}}{\mathrm{c}} \\ $$
Question Number 151069 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}\:\left(\mathrm{1145141919810893x}\right)}{\mathrm{x}\left(\mathrm{cosh}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)}\mathrm{dx}=\frac{\pi}{\mathrm{4}} \\ $$
Question Number 151068 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\int_{\mathrm{0}} ^{\infty} \frac{\sqrt{\mathrm{x}}}{\left(\mathrm{x}^{\mathrm{4}} +\mathrm{14x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{5}}{\mathrm{4}}} }\mathrm{dx}=\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\mathrm{4}\sqrt{\mathrm{2}\pi}} \\ $$
Question Number 151067 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\pi} \mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{2}}\centerdot\mathrm{arctan}\left(\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{x}}−\mathrm{1}\right)\mathrm{dx}=\sqrt{\mathrm{2}}\mathrm{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)+\left(\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)\pi \\ $$
Question Number 151066 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left(\mathrm{arctan}\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{2}}\right)+\mathrm{arctan}\left(\frac{\mathrm{cos}\:\mathrm{3x}+\mathrm{15cos}\:\mathrm{x}}{\mathrm{8}}\right)\right)\mathrm{dx} \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\mathrm{2}}\right) \\ $$
Question Number 151065 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:\:::\:\:\int_{−\infty} ^{+\infty} \frac{\Gamma\left(\mathrm{x}\right)}{\Gamma\left(\mathrm{x}+\mathrm{a}\right)}\mathrm{sin}\:\left(\pi\mathrm{x}\right)\mathrm{dx}=\frac{\mathrm{2}^{\mathrm{a}−\mathrm{1}} }{\Gamma\left(\mathrm{a}\right)}\pi\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{a}>\mathrm{0}\right) \\ $$
Question Number 151064 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\pi} \mathrm{arctan}\left(\frac{\mathrm{2sin}\:^{\mathrm{2}} \mathrm{x}}{\mathrm{1}−\mathrm{2}\sqrt{\mathrm{2}}\varphi\mathrm{cos}\:\mathrm{x}+\mathrm{2}\varphi^{\mathrm{2}} }\right)\mathrm{dx}=\pi\mathrm{arctan}\:\sqrt{\varphi}\:\:\:\:\:\:\:\:\:\:\left(\varphi=\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{2}}\right) \\ $$
Question Number 151063 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\left(\pi^{\mathrm{2}} +\mathrm{ln}^{\mathrm{2}} \mathrm{x}\right)\mathrm{x}}\mathrm{dx}=\gamma \\ $$
Question Number 151062 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\centerdot\sqrt[{\mathrm{4}}]{\mathrm{8x}^{\mathrm{2}} +\mathrm{8x}+\mathrm{1}}}=\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)}{\mathrm{2}^{\frac{\mathrm{11}}{\mathrm{4}}} \Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$
Question Number 151061 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:\:::\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{x}\centerdot\mathrm{cot}\:\mathrm{x}\centerdot\mathrm{ln}^{\mathrm{2}} \mathrm{cos}\:\mathrm{xdx}=\frac{\pi^{\mathrm{3}} }{\mathrm{24}}\mathrm{ln2}+\frac{\pi}{\mathrm{6}}\mathrm{ln}^{\mathrm{3}} \mathrm{2}−\frac{\mathrm{3}}{\mathrm{16}}\pi\zeta\left(\mathrm{3}\right) \\ $$
Question Number 151060 Answers: 0 Comments: 0
$$\mathrm{Calculate}\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}\sqrt{\mathrm{x}}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\mathrm{ax}\right)}\mathrm{dx}=\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{a}+\sqrt{\mathrm{2a}}}{\:\sqrt{\mathrm{2}}\mathrm{a}\left(\mathrm{1}+\mathrm{a}^{\mathrm{2}} \right)}\pi\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:,\left(\mathrm{a}>\mathrm{0}\right) \\ $$
Question Number 151059 Answers: 0 Comments: 0
$$\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}\:+\mathrm{2}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{2}−\sqrt[{\mathrm{3}}]{\mathrm{x}+\mathrm{2}}}\:=\:\mathrm{2}\: \\ $$$$\:\:\:\:\mathrm{x}\:=? \\ $$
Question Number 151052 Answers: 1 Comments: 0
$$\mathrm{if}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c}\:\:\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\mathrm{and} \\ $$$$\frac{\mathrm{a}}{\mathrm{1}+\mathrm{a}}\:+\:\frac{\mathrm{b}}{\mathrm{1}+\mathrm{b}}\:+\:\frac{\mathrm{c}}{\mathrm{1}+\mathrm{c}}\:=\:\mathrm{1}\:\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{abc}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$
Question Number 151045 Answers: 0 Comments: 0
Question Number 151043 Answers: 2 Comments: 0
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}: \\ $$$$\begin{cases}{\boldsymbol{\mathrm{y}}\:=\:\frac{\mathrm{2x}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\\{\boldsymbol{\mathrm{z}}\:=\:\frac{\mathrm{2y}}{\mathrm{1}−\mathrm{y}^{\mathrm{2}} }}\\{\boldsymbol{\mathrm{x}}\:=\:\frac{\mathrm{2z}}{\mathrm{1}−\mathrm{z}^{\mathrm{2}} }}\end{cases} \\ $$
Question Number 151042 Answers: 0 Comments: 1
$$\mathrm{if}\:\:\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\mathrm{1}}\:+\:\sqrt{\sqrt[{\mathrm{3}}]{\mathrm{16}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{4}}}\:=\:\sqrt{\boldsymbol{\mathrm{x}}}\:\:;\:\:\boldsymbol{\mathrm{x}}\in\mathbb{Z} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$
Question Number 151037 Answers: 1 Comments: 0
$$\underset{\mathrm{r}=\mathrm{1},\mathrm{r}\neq\mathrm{s}} {\overset{\mathrm{n}} {\sum}}\:\:\underset{\mathrm{s}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{rs}}{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)}\overset{?} {=}\frac{\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{3n}+\mathrm{2}\right)}{\mathrm{12}} \\ $$
Question Number 151036 Answers: 0 Comments: 0
$$ \\ $$
Question Number 151034 Answers: 0 Comments: 0
Question Number 151035 Answers: 0 Comments: 0
$$\begin{vmatrix}{}&{}\\{}&{}\end{vmatrix} \\ $$
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