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Category: Integration

Question-96637

Question Number 96637 by 175 last updated on 03/Jun/20 Commented by bemath last updated on 03/Jun/20 $$\int\:\frac{\mathrm{ln}\left(\mathrm{x}^{\mathrm{a}} \right)}{\mathrm{x}}\:\mathrm{dx}\:=\:\mathrm{a}\int\:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{dx} \\ $$$$\mathrm{let}\:\mathrm{u}\:=\:\mathrm{ln}\:\left(\mathrm{x}\right)\:\Rightarrow\:\mathrm{du}\:=\:\frac{\mathrm{dx}}{\mathrm{x}} \\ $$$$\Rightarrow\mathrm{a}\int\:\mathrm{u}\:\mathrm{du}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{a}\:\mathrm{u}^{\mathrm{2}} +\mathrm{c}\: \\ $$$$=\:\frac{\mathrm{a}}{\mathrm{2}}\:\left(\mathrm{ln}\left(\mathrm{x}\right)\right)^{\mathrm{2}}…

x-2-4x-1-0-are-roots-3-17-5-solution-is-root-2-4-1-0-2-4-1-3-17-5-2-17-5-

Question Number 162174 by mnjuly1970 last updated on 27/Dec/21 $$ \\ $$$$\:\:\:\:{x}^{\:\mathrm{2}} −\:\mathrm{4}{x}\:−\mathrm{1}=\mathrm{0}\:\: \\ $$$$\:\:\:\:\:\alpha\:,\:\beta\:\:{are}\:{roots}\: \\ $$$$\:\:\:\:\:\alpha^{\:\mathrm{3}} \:+\:\mathrm{17}\beta\:+\mathrm{5}\:=? \\ $$$$\:\:−−−{solution}−−− \\ $$$$\:\:\:\alpha\:\:\:{is}\:{root}\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}} −\mathrm{4}\alpha\:−\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}}…