Integral of $\cot(x)$

by Jakub Marian

Tip: See my list of the Most Common Mistakes in English. It will teach you how to avoid mis­takes with com­mas, pre­pos­i­tions, ir­reg­u­lar verbs, and much more (PDF Version).

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 $$ ∫ \frac{\cos(x)}{\sin(x)}\,dx = ∫ \frac{1}{u}\,du = \log(u)+C = \log(\sin(x))+C $$

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