To integrate the function \(\cot(x + a)\) with respect to \(x\), we can follow these steps:

### Problem:
\[
\int \cot(x + a) \, dx
\]

### Solution:

We know that \(\cot(x + a)\) can be written as:
\[
\cot(x + a) = \frac{\cos(x + a)}{\sin(x + a)}
\]
This suggests a natural substitution.

Let:
\[
u = \sin(x + a)
\]
Then, the derivative of \(u\) with respect to \(x\) is:
\[
du = \cos(x + a) \, dx
\]
Thus, the integral becomes:
\[
\int \frac{\cos(x + a)}{\sin(x + a)} \, dx = \int \frac{du}{u}
\]
The integral of \(\frac{1}{u}\) is:
\[
\ln|u| + C
\]
Substituting back \(u = \sin(x + a)\), we get:
\[
\int \cot(x + a) \, dx = \ln|\sin(x + a)| + C
\]

### Final Answer:
\[
\int \cot(x + a) \, dx = \ln|\sin(x + a)| + C
\]

Let me know if you'd like further details or clarification!

Here are 5 related questions to expand on this:

1. How would you integrate \(\csc(x + a)\)?
2. What is the integral of \(\tan(x + a)\)?
3. How do you handle the integral of trigonometric functions combined with exponentials, like \(\int e^x \cot(x + a) \, dx\)?
4. How do you apply integration by parts to functions involving \(\cot(x + a)\)?
5. What happens if \(a\) is a specific constant, such as 0?

**Tip**: Substitution is a common method for handling integrals involving trigonometric functions. Look for expressions that simplify using standard trigonometric identities.

Learn how to solve the integral of the cotangent function ∫cot(x + a) dx using substitution. This step-by-step solution covers key trigonometric integration techniques and calculus concepts, suitable for high school students in Grades 11-12.

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