To solve the integral of $\cos(2x) \ln(3 + 4\tan(x))$, we need to apply integration techniques involving both trigonometric and logarithmic functions. Let's denote the integral as:

$I = \int \cos(2x) \ln(3 + 4\tan(x)) \, dx$

This integral does not appear to have a straightforward solution using elementary functions, but we can attempt to make progress by using substitutions or integration by parts.

Step 1: Consider a possible substitution.

Since $\cos(2x)$ and $\tan(x)$ are related through trigonometric identities, we might explore the substitution: $u = \tan(x)$ This gives us: $du = \sec^2(x) \, dx$ and $\cos(2x) = \frac{1 - \tan^2(x)}{1 + \tan^2(x)} = \frac{1 - u^2}{1 + u^2}$

This substitution transforms the integral, but due to the complexity introduced by the logarithmic function $\ln(3 + 4u)$, it doesn't immediately simplify the problem. A direct application of integration by parts might be more useful.

Step 2: Integration by parts.

We can attempt to integrate by parts using the formula: $\int u \, dv = uv - \int v \, du$

Let:

• $u = \ln(3 + 4\tan(x))$, hence $du = \frac{4 \sec^2(x)}{3 + 4\tan(x)} \, dx$
• $dv = \cos(2x) \, dx$, hence $v = \frac{1}{2} \sin(2x)$

The integral becomes: $I = \frac{1}{2} \sin(2x) \ln(3 + 4\tan(x)) - \frac{1}{2} \int \sin(2x) \cdot \frac{4 \sec^2(x)}{3 + 4\tan(x)} \, dx$

The second integral on the right-hand side still seems quite complicated, involving both $\sin(2x)$ and $\sec^2(x)$ in the numerator. This suggests the integral might not have a simple closed form in elementary functions.

Conclusion:

The integral of $\cos(2x) \ln(3 + 4\tan(x))$ likely does not have a simple elementary solution, and it may require advanced techniques or numerical methods to evaluate. It is also possible that this integral could be expressed in terms of special functions.

Would you like more details on a specific step, or further exploration on how to numerically evaluate it?

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Here are 5 related questions that expand on this information:

1. How do you apply the method of integration by parts to functions involving trigonometric and logarithmic terms?
2. What are special functions, and how can they help in solving integrals that lack elementary solutions?
3. How can numerical methods be used to approximate difficult integrals?
4. What is the trigonometric identity for $\cos(2x)$ and how is it derived?
5. How does substitution simplify certain types of integrals?

Tip: When dealing with integrals involving both trigonometric and logarithmic functions, try simplifying the integral with substitutions related to the trigonometric identities before applying other methods like integration by parts.