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.