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Question-96637




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

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