What is the integral of cot(x)?

integral of cot(x)

The integral of

\cot(x)

\ln \lvert \sin(x) \rvert + C

.

Proof. By definition, we have that

\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}

. So:

\begin{align*} \int \cot(x) dx = \int \frac{1}{\tan(x)} dx = \int \frac{\cos(x)}{\sin(x)} dx. \end{align*}

Now we want to apply the substitution method. Let

u = \sin(x)

. Then we get:

\begin{align*} \frac{d}{dx} u = \cos(x) \iff du = \cos(x)dx. \end{align*}

Together, we have that:

\begin{align*} \int \cot(x) dx = \int \frac{1}{\tan(x)} dx &= \int \frac{\cos(x)}{\sin(x)} dx \\ &= \int \frac{1}{u} du \\ &= \ln \lvert u \rvert + C \\ &= \ln \lvert \sin(x) \rvert + C \quad \text{since } u = \sin(x). \end{align*}

Therefore, the integral of

\cot(x)

\ln \lvert \sin(x) \rvert + C