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The exercise asks to find the antiderivatives (or primitives) of the given functions. Let's go through some of them:

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

To solve this, we can use a substitution:
Let \( u = 1 + e^{3x} \), then \( du = 3e^{3x} dx \). The result will lead to an expression involving the natural logarithm.

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

This is a standard form. We know that the antiderivative of \( \frac{\ln(x)}{x} \) is \( \frac{\ln^2(x)}{2} \), because by applying integration by parts, we can solve this.

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

This can be simplified using the identity \( \sin^2(x) = 1 - \cos^2(x) \), and then solved by basic trigonometric integration techniques.

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

This requires the substitution \( u = \ln(x) \), leading to a result involving the natural logarithm.

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Would you like me to continue with specific steps for any of the functions or clarify one in detail? Here are 5 follow-up questions to extend the current discussion:

1. Would you like a detailed step-by-step solution for problem 1?
2. Do you want to explore the method of substitution in more detail?
3. Should we focus on a specific integration technique like parts or trigonometric identities?
4. Are there any particular integrals from the list you'd like prioritized?
5. Would it be helpful to review general strategies for tackling integrals?

**Tip:** Always try substitution first when dealing with logarithmic and exponential functions.

Déterminer toutes les primitives des fonctions suivantes : f(x) = e^(3x) / (1 + e^(3x)), f(x) = ln(x)/x, f(x) = cos(x)sin^2(x), etc.

Step-by-step solutions for finding the antiderivatives of various functions, including e^(3x)/(1+e^(3x)), ln(x)/x, cos(x)sin^2(x), and more. This covers techniques like substitution, trigonometric identities, and integration by parts. Suitable for undergraduate calculus students.

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