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Question Number 125965 by bramlexs22 last updated on 16/Dec/20

    ∫ cot x ln (sin x) dx ?

$$\:\:\:\:\int\:\mathrm{cot}\:{x}\:\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)\:{dx}\:? \\ $$

Answered by Lordose last updated on 16/Dec/20

  ∫cot(x)ln(sin(x))dx =^(u=ln(sin(x))) ∫udu     = (u^2 /2) + C = ((ln^2 sin(x))/2) + C

$$ \\ $$$$\int\mathrm{cot}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{sin}\left(\mathrm{x}\right)\right)\mathrm{dx}\:\overset{\mathrm{u}=\mathrm{ln}\left(\mathrm{sin}\left(\mathrm{x}\right)\right)} {=}\int\mathrm{udu}\:\:\: \\ $$$$=\:\frac{\mathrm{u}^{\mathrm{2}} }{\mathrm{2}}\:+\:\mathrm{C}\:=\:\frac{\mathrm{ln}^{\mathrm{2}} \mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{2}}\:+\:\mathrm{C} \\ $$

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