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Let's go through the list and determine the primitives (antiderivatives) of the functions you provided. Here's the process for each one:

### 1. \( f(x) = \frac{e^{3x}}{1 + e^{3x}} \)

For this type of function, substitution works best. Set \( u = 1 + e^{3x} \), which makes \( du = 3e^{3x} dx \), and proceed with integration.

### 2. \( f(x) = \frac{\ln(x)}{x} \)

This is a common form. The antiderivative of \( \frac{\ln(x)}{x} \) is \( \frac{(\ln(x))^2}{2} + C \), where \( C \) is the constant of integration.

### 3. \( f(x) = \cos(x) \sin^2(x) \)

We can simplify using a trigonometric identity or substitution. Rewriting \( \sin^2(x) \) in terms of \( 1 - \cos^2(x) \) might help, or using a substitution like \( u = \sin(x) \).

### 4. \( f(x) = \frac{1}{x \ln(x)} \)

The antiderivative of \( \frac{1}{x \ln(x)} \) is \( \ln|\ln(x)| + C \).

### 5. \( f(x) = 3x \sqrt{1 + x^2} \)

This can be solved using a trigonometric substitution, such as \( x = \sinh(t) \), or by recognizing a pattern for the derivative of \( x^2 \).

### 6. \( f(x) = (3x - 1)(3x^2 - 2x + 3)^3 \)

Here, you can apply a substitution like \( u = 3x^2 - 2x + 3 \) and proceed by integrating the polynomial form.

### 7. \( f(x) = \frac{1 - x^2}{(x^3 - 3x + 1)^3} \)

This would involve a complex substitution or possibly partial fraction decomposition depending on the nature of the denominator.

### 8. \( f(x) = \frac{x - 1}{\sqrt{x(x - 2)}} \)

A substitution, such as \( u = x(x - 2) \), will help simplify the expression for integration.

### 9. \( f(x) = \frac{1}{x \ln(x^2)} \)

Simplifying \( \ln(x^2) = 2\ln(x) \), the integral becomes \( \frac{1}{2x \ln(x)} \), whose primitive is \( \frac{\ln|\ln(x)|}{2} + C \).

### 10. \( f(x) = \arctan(x) \)

The antiderivative of \( \arctan(x) \) is \( x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C \).

### 11. \( f(x) = (\ln(x))^2 \)

Here we use integration by parts. Let \( u = (\ln(x))^2 \) and \( dv = dx \). After differentiating and integrating, combine the results.

### 12. \( f(x) = \sin(\ln(x)) \)

Substitution \( u = \ln(x) \) reduces this to \( \sin(u) \), whose antiderivative is \( -\cos(u) + C = -\cos(\ln(x)) + C \).

### 13. \( f(x) = \sin^5(x) \)

Use the identity \( \sin^5(x) = (\sin(x))^5 \), and possibly reduce powers of sine using known reduction formulas.

### 14. \( f(x) = \frac{1}{\cos^6(x)} \) on \( \left] -\frac{\pi}{2}, \frac{\pi}{2} \right[ \)

This is \( \sec^6(x) \), which has a standard antiderivative involving multiple terms of secants and tangents.

### 15. \( \cos^4(x) \sin^2(x) \)

This can be simplified using the identity for powers of sine and cosine or substitution like \( u = \cos(x) \).

### 16. \( \cos^3(x) \cos(3x) \)

Using product-to-sum formulas or a substitution like \( u = \cos(3x) \) will simplify the integral.

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Would you like to focus on a specific question from these, or do you want detailed solutions for any particular problem?

### Follow-up Questions:
1. How do trigonometric identities help in simplifying integrals?
2. What is the general approach for solving integrals involving natural logarithms?
3. How can substitution simplify complex algebraic expressions in integration?
4. How does integration by parts work, and when should it be applied?
5. Why are powers of sine and cosine often simplified using reduction formulas?

### Tip:
For integrals involving complex expressions or products of functions, substitution is often the most efficient way to simplify the integration process.

Determine the antiderivatives of the given functions starting from question 10.

Find the antiderivatives of functions such as arctan(x), (ln(x))^2, and sin(ln(x)) using integration techniques like substitution and parts. This solution covers trigonometric and logarithmic integrals, providing a detailed step-by-step approach for each problem.

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