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To solve the integral \( I = \int \frac{\cos^3 x}{1 + \sin x} \, dx \), we can use a substitution method. 

### Solution:

1. **Substitute** \( u = \sin x \):
   - Then \( du = \cos x \, dx \).
   - Rewrite the integral in terms of \( u \).

Since \( \cos^2 x = 1 - \sin^2 x \), we can express \( \cos^3 x \) as \( \cos^2 x \cdot \cos x = (1 - \sin^2 x) \cdot \cos x \).

Therefore:
\[ \cos^3 x \, dx = (1 - u^2) \, du \]

So the integral becomes:
\[ I = \int \frac{(1 - u^2) \, du}{1 + u} \]

2. **Simplify the Integrand**:
   - Split the fraction:
     \[ \frac{1 - u^2}{1 + u} = \frac{1 - u^2}{1 + u} = \frac{(1 - u)(1 + u)}{1 + u} \]
   - Simplify:
     \[ \frac{(1 - u)(1 + u)}{1 + u} = 1 - u \]

3. **Integrate**:
   \[ I = \int (1 - u) \, du \]
   - Integrate term by term:
     \[ I = \int 1 \, du - \int u \, du \]
     \[ I = u - \frac{u^2}{2} + C \]

4. **Substitute back \( u = \sin x \)**:
   \[ I = \sin x - \frac{\sin^2 x}{2} + C \]

Therefore, the solution to the integral is:
\[ I = \sin x - \frac{\sin^2 x}{2} + C \]

Would you like more details on any part of this solution, or do you have any specific questions?

### Further Questions
1. How do you find the integral of more complex trigonometric functions?
2. What is the general approach for solving integrals using substitution?
3. How does the integration technique change if the integrand includes both sine and cosine terms in different forms?
4. Can you explain the method of partial fractions in integration?
5. How would you handle definite integrals with trigonometric functions?
6. What are some common trigonometric identities useful in integration?
7. How can you verify the result of an integral solution?
8. What are some applications of trigonometric integrals in real-world problems?

### Tip:
When solving integrals involving trigonometric functions, identifying useful substitutions and trigonometric identities can significantly simplify the problem.

Learn how to solve the integral of cos^3(x) / (1 + sin(x)) step-by-step using the substitution method. This problem covers integral calculus and trigonometric integrals, focusing on the substitution technique. Suitable for advanced high school and college-level students.

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