$$\begin{align*} \boxed{\int \cot(x) \, dx = \ln|\sin(x)| + C} \end{align*}$$

where $C$ is the constant of integration.

Step-by-Step Solution:

In this article, we’ll go through the steps to find the integral of $\cot(x)$. The function $\cot(x)$ is the ratio of $\cos(x)$ to $\sin(x)$, which is a more convenient form when integrating.

Since $\cot(x) = \dfrac{\cos(x)}{\sin(x)}$:

$$\begin{align*} \int \cot(x) \, dx = \int \dfrac{\cos(x)}{\sin(x)} \, dx \end{align*}$$

To integrate this, we can use substitution. Let:

$$\begin{align*} u = \sin(x) \end{align*}$$

Now, differentiate $u$ with respect to $x$:

$$\begin{align*} \dfrac{du}{dx} = \cos(x) \Rightarrow dx = \dfrac{du}{\cos(x)} \end{align*}$$

Substituting $u$ and $dx$:

$$\begin{align*} \int \dfrac{\cos(x)}{\sin(x)} \, dx = \int \dfrac{1}{u} \, du \end{align*}$$

The integral of $\dfrac{1}{u}$ is $\ln|u|$, so after substituting back in $x$, we have:

$$\begin{align*} \boxed{\int \cot(x) \, dx = \ln|\sin(x)| + C} \end{align*}$$

Why This Works:

By rewriting $\cot(x)$ as $\dfrac{\cos(x)}{\sin(x)}$, we could see that using substitution was a straightforward way to simplify the integration process. It is very common to break down trigonometric functions into cosine and sine when possible!

Final Answer:

To summarize, the integral of $\cot(x)$ is:

$$\begin{align*} \int \cot(x) \, dx = \ln|\sin(x)| + C \end{align*}$$

This result is a standard problem for anyone studying integral Calculus with trigonometric functions!