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The image contains steps for solving an integral:

\[
\int_0^{\frac{\pi}{4}} \frac{dx}{3 + \cos(2x)}
\]

Here is a breakdown of the solution steps:

1. **Initial Integral Setup**:
   The integral is transformed using a trigonometric identity for \( \cos(2x) \):
   \[
   \cos(2x) = 1 - 2\sin^2(x)
   \]
   The equation becomes:
   \[
   \int_0^{\frac{\pi}{4}} \frac{dx}{3 + \cos(2x)} = \int_0^{\frac{\pi}{4}} \frac{dx}{3 + (1 - 2\sin^2(x))}
   \]
   Simplifying:
   \[
   \int_0^{\frac{\pi}{4}} \frac{dx}{4 - 2\sin^2(x)}
   \]

2. **Simplification**:
   By rewriting the denominator using trigonometric identities:
   \[
   4 - 2\sin^2(x) = 2(2 - \sin^2(x)) = 2\cos^2(x)
   \]
   The integral becomes:
   \[
   \frac{1}{2} \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^2(x) \left(2 + \tan^2(x)\right)}
   \]

3. **Substitution**:
   Using the substitution \( t = \tan(x) \), where \( dt = \sec^2(x) dx \), the integral transforms into:
   \[
   \frac{1}{2} \int_0^1 \frac{dt}{2 + t^2}
   \]

4. **Standard Integral**:
   The integral \( \int \frac{dt}{a + t^2} \) is a standard form, which results in:
   \[
   \frac{1}{a} \arctan\left(\frac{t}{\sqrt{a}}\right)
   \]
   Applying this to the current integral:
   \[
   \frac{1}{2} \times \frac{1}{\sqrt{2}} \arctan\left(\frac{t}{\sqrt{2}}\right) \Bigg|_0^1
   \]

5. **Final Result**:
   Substituting the limits of integration \( t = 1 \) and \( t = 0 \):
   \[
   \frac{1}{2\sqrt{2}} \left( \arctan\left(\frac{1}{\sqrt{2}}\right) - \arctan(0) \right)
   \]
   The final answer involves evaluating this expression.

---

Would you like more details on any step, or assistance with specific parts of the calculation? Here are some relative questions to consider:

1. How is the trigonometric identity for \( \cos(2x) \) applied in integrals?
2. What is the significance of the substitution \( t = \tan(x) \)?
3. How do you evaluate standard integrals involving \( \frac{1}{a + t^2} \)?
4. What are the steps to compute \( \arctan \) in definite integrals?
5. What strategies simplify integrals involving trigonometric functions?

**Tip**: Substitutions, particularly trigonometric ones, are a powerful tool to simplify integrals involving complex expressions.

Solve the integral \( \int_0^{\frac{\pi}{4}} \frac{dx}{3 + \cos(2x)} \)

Learn how to solve the definite integral \( \int_0^{\frac{\pi}{4}} \frac{dx}{3 + \cos(2x)} \) using trigonometric identities and substitution techniques. Explore step-by-step breakdowns for transforming the integral, simplifying the expression, and applying standard integral forms involving arctangents. Suitable for advanced calculus students in Grades 11-12.

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