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To solve the given definite integral using U-substitution, let's follow these steps:

### Problem
Evaluate the following integral using U-substitution:
\[
\int_0^{\frac{\pi}{4}} \frac{\sin(2x)}{1 + \cos(2x)} \, dx
\]

### Solution
1. **Substitute:**  
   Let \( u = 1 + \cos(2x) \).

2. **Differentiate:**  
   Find \( du \).  
   Since \(\frac{d}{dx} \cos(2x) = -2 \sin(2x)\), we have:
   \[
   du = -2 \sin(2x) \, dx \implies \sin(2x) \, dx = -\frac{1}{2} \, du
   \]

3. **Change Limits:**  
   When \( x = 0 \):  
   \[
   u = 1 + \cos(0) = 2
   \]  
   When \( x = \frac{\pi}{4} \):  
   \[
   u = 1 + \cos\left(2 \times \frac{\pi}{4}\right) = 1 + \cos\left(\frac{\pi}{2}\right) = 1
   \]

4. **Rewrite the Integral:**  
   Substitute into the integral:
   \[
   \int_{2}^{1} \frac{\sin(2x)}{1 + \cos(2x)} \, dx = \int_{2}^{1} \frac{-\frac{1}{2} \, du}{u}
   \]

5. **Simplify and Integrate:**  
   \[
   = -\frac{1}{2} \int_{2}^{1} \frac{1}{u} \, du = -\frac{1}{2} \left[ \ln |u| \right]_{2}^{1}
   \]

6. **Evaluate the Definite Integral:**
   \[
   -\frac{1}{2} \left( \ln(1) - \ln(2) \right) = -\frac{1}{2} (0 - \ln(2)) = \frac{\ln(2)}{2}
   \]

### Final Answer
\[
\int_0^{\frac{\pi}{4}} \frac{\sin(2x)}{1 + \cos(2x)} \, dx = \frac{\ln(2)}{2}
\]

Would you like more details or have any questions about the solution?

### Related Questions
1. How do you choose the right substitution for an integral?
2. What are the common techniques for integrating trigonometric functions?
3. Can U-substitution be used for indefinite integrals as well?
4. How do you handle integrals that involve squared trigonometric functions?
5. What are some alternative methods for evaluating this integral?

### Tip
Always check the new limits of integration after substitution, as they correspond to the new variable \(u\).

Evaluate the following integral using U-substitution: ∫₀^(π/4) [sin(2x) / (1 + cos(2x))] dx

This solution explains how to solve the definite integral ∫₀^(π/4) [sin(2x) / (1 + cos(2x))] dx using U-substitution. Follow step-by-step through the substitution of u = 1 + cos(2x) and the integration process. Learn about trigonometric integration techniques and the Fundamental Theorem of Calculus.

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