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To integrate $\cot(x)$, recall that
$$ \cot(x) = \frac{\cos(x)}{\sin(x)}\,, $$so
$$ ∫ \cot(x)\,dx = ∫ \frac{\cos(x)}{\sin(x)}\,dx\,. $$ By choosing $u = \sin(x)$, that is, “$du = \cos(x)\,dx$” (in quotation marks because this expression does not make sense mathematically, but it does work formally), we get $$ ∫ \frac{\cos(x)}{\sin(x)}\,dx = ∫ \frac{1}{u}\,du = \log(u)+C = \log(\sin(x))+C $$By the way, have you already seen my brand new web app for non-native speakers of English? It's based on reading texts and learning by having all meanings, pronunciations, grammar forms etc. easily accessible. It looks like this:
