The integral given in the image is:

$\int \frac{1}{1 + \cos(2x)} \, dx$

To solve this, we can simplify the problem using a trigonometric identity and substitution.

Steps to Solve:

1. Use the trigonometric identity for $\cos(2x)$: $\cos(2x) = 2\cos^2(x) - 1 \implies 1 + \cos(2x) = 2\cos^2(x)$ So the integral becomes: $\int \frac{1}{1 + \cos(2x)} \, dx = \int \frac{1}{2\cos^2(x)} \, dx = \frac{1}{2} \int \sec^2(x) \, dx$

2. Integrate $\sec^2(x)$: The integral of $\sec^2(x)$ is: $\int \sec^2(x) \, dx = \tan(x)$

3. Combine the result: $\frac{1}{2} \int \sec^2(x) \, dx = \frac{1}{2} \tan(x) + C$

Thus, the solution is: $\boxed{\frac{1}{2} \tan(x) + C}$

Final Answer:

$\frac{1}{2} \tan(x) + C$

Let me know if you'd like me to elaborate further or if you have questions! Here are 5 related problems to deepen your understanding:

1. Solve $\int \sec^2(x) \, dx$.
2. Simplify and solve $\int \frac{1}{1 - \cos(2x)} \, dx$.
3. Derive the identity $\cos(2x) = 2\cos^2(x) - 1$ and use it in another integral.
4. Solve $\int \frac{1}{\sin^2(x)} \, dx$.
5. Explain why substitution wasn't required in this problem.

Tip: Memorize the common trigonometric integrals like $\int \sec^2(x) \, dx = \tan(x)$ to solve problems faster!