Understanding the Integral of cot(x)

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

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!