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The integral given is:

\[
\int \frac{\sec x}{\sec x + \tan x} \, dx
\]

To solve this integral, we can proceed by using substitution.

### Solution:

1. **Substitute** \( u = \sec x + \tan x \).

2. Differentiate \( u \) with respect to \( x \):
   \[
   \frac{du}{dx} = \sec x \tan x + \sec^2 x
   \]
   This simplifies to:
   \[
   du = (\sec x \tan x + \sec^2 x) \, dx
   \]

3. Notice that \(\sec x \, dx\) appears in the integral, and we can rewrite:
   \[
   \sec x \, dx = \frac{du}{\sec x \tan x + \sec^2 x} = \frac{du}{\sec x + \tan x} = \frac{du}{u}
   \]

4. Now the integral becomes:
   \[
   \int \frac{1}{u} \, du
   \]

5. Integrating this, we get:
   \[
   \ln |u| + C
   \]

6. Substitute back for \( u \):
   \[
   \ln |\sec x + \tan x| + C
   \]

### Final Answer:
\[
\int \frac{\sec x}{\sec x + \tan x} \, dx = \ln |\sec x + \tan x| + C
\]

Would you like more details on any step?

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Here are some related questions for further exploration:

1. How do we choose substitutions effectively in integrals?
2. Why does \( u = \sec x + \tan x \) simplify this integral?
3. Can this method be used for other trigonometric integrals?
4. How would the solution change if the integral was \( \int \frac{\sec^2 x}{\sec x + \tan x} \, dx \)?
5. What are the applications of integrals involving \( \sec x \) and \( \tan x \)?

**Tip:** When facing integrals with expressions like \( \sec x + \tan x \) in the denominator, consider substitutions based on known derivatives, as it often simplifies the integration process.

Evaluate the integral \( \int \frac{\sec x}{\sec x + \tan x} \, dx \).

This page provides a detailed solution to the integral \( \int \frac{\sec x}{\sec x + \tan x} \, dx \) using substitution. Ideal for undergraduate calculus students, the solution explores substitution methods and trigonometric identities.

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