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Question-203675

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Question Number 203675 by 073 last updated on 25/Jan/24 Answered by Frix last updated on 25/Jan/24 $${x}^{\mathrm{4}} −\mathrm{2}{ax}^{\mathrm{2}} −{x}+{a}^{\mathrm{2}} −{a}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} +{x}−{a}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}−{a}\right)=\mathrm{0} \\…

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Question-203668

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Question Number 203668 by Perelman last updated on 25/Jan/24 Answered by witcher3 last updated on 25/Jan/24 $$\mathrm{x}\overset{\mathrm{f}} {\rightarrow}\frac{\mathrm{1}}{\mathrm{4}−\mathrm{3x}} \\ $$$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{3}}{\left(\mathrm{4}−\mathrm{3x}\right)^{\mathrm{2}} } \\ $$$$\forall\mathrm{x}\in\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\in\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…..\left(\mathrm{1}\right) \\ $$$$\left.\Rightarrow\mathrm{f}\left[−\mathrm{1},\frac{\mathrm{1}}{\mathrm{3}}\right]\right)\subset\left[\frac{\mathrm{1}}{\mathrm{7}},\frac{\mathrm{1}}{\mathrm{3}}\right]…

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1-oo-1-x-x-2-x-1-dx-

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Question Number 203681 by SANOGO last updated on 25/Jan/24 $$\int_{\mathrm{1}} ^{+{oo}} \frac{\mathrm{1}}{{x}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$ Answered by Frix last updated on 25/Jan/24 $${t}=\frac{\mathrm{2}{x}+\mathrm{1}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{\:\sqrt{\mathrm{3}}} \\…

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