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