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