The integral we are dealing with is:

\[
\int \sqrt{3 + \cos(2x)} \, dx
\]

This is a non-trivial integral that cannot be solved by elementary methods directly. However, the general approach to this type of problem often involves trigonometric identities, substitution, or looking into elliptic integrals. 

### Step-by-step approach:

1. **Rewrite \( \cos(2x) \) using a trigonometric identity**:
   \[
   \cos(2x) = 2\cos^2(x) - 1
   \]
   So,
   \[
   3 + \cos(2x) = 3 + (2\cos^2(x) - 1) = 2\cos^2(x) + 2
   \]
   Hence, the integral becomes:
   \[
   \int \sqrt{2(\cos^2(x) + 1)} \, dx = \int \sqrt{2} \sqrt{\cos^2(x) + 1} \, dx
   \]
   Simplifying further:
   \[
   \sqrt{2} \int \sqrt{\cos^2(x) + 1} \, dx
   \]

2. **Handle \( \cos^2(x) + 1 \)**:
   While the integral of \( \sqrt{\cos^2(x) + 1} \) doesn't have a simple closed-form in terms of elementary functions, you can use a substitution involving elliptic integrals. This problem leads to elliptic integral solutions due to the form of the expression.

   Therefore, the integral is expressed as:

\[
\int \sqrt{3 + \cos(2x)} \, dx = \text{elliptic integral form}.
\]

### Conclusion:
This integral doesn't have a simple elementary solution, but it can be expressed using elliptic integrals. If you are familiar with elliptic integrals, you can further explore that path.

Would you like more details on elliptic integrals or further explanation on handling this kind of integral?

### Related questions:
1. What are elliptic integrals, and how do they apply to this problem?
2. How can trigonometric identities simplify complex integrals?
3. Are there cases where numerical integration is preferred over analytical methods?
4. What are some standard methods for solving integrals involving square roots of trigonometric functions?
5. How do substitutions like \( u = \sin(x) \) or \( u = \cos(x) \) help solve integrals?

### Tip:
For integrals involving square roots and trigonometric functions, always check for identities or substitutions that might reduce the complexity.

Discover how to solve the integral of sqrt(3 + cos(2x)) using trigonometric identities and elliptic integrals. This problem provides insights into advanced calculus techniques and is suitable for college-level students exploring non-trivial integrals.

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