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The problem in the image asks to prove in two ways that the function \( f : (1, +\infty) \to \mathbb{R}, \ f(x) = \ln(1 + \ln x) \) is a primitive (antiderivative) of the function \( g : (1, +\infty) \to \mathbb{R}, \ g(x) = \frac{1}{x(1 + \ln x)} \).

### Method 1: Direct differentiation

To show that \( f(x) = \ln(1 + \ln x) \) is a primitive of \( g(x) = \frac{1}{x(1 + \ln x)} \), we need to differentiate \( f(x) \) and check if we obtain \( g(x) \).

#### Step 1: Differentiate \( f(x) = \ln(1 + \ln x) \)
Using the chain rule, we differentiate \( f(x) \) as follows:
\[
f'(x) = \frac{d}{dx} \ln(1 + \ln x)
\]
The derivative of \( \ln(u) \) is \( \frac{1}{u} \), so we get:
\[
f'(x) = \frac{1}{1 + \ln x} \cdot \frac{d}{dx}(\ln x)
\]
Now, the derivative of \( \ln x \) is \( \frac{1}{x} \), so we have:
\[
f'(x) = \frac{1}{1 + \ln x} \cdot \frac{1}{x}
\]
Thus, 
\[
f'(x) = \frac{1}{x(1 + \ln x)}
\]
which is exactly \( g(x) \).

Hence, \( f(x) = \ln(1 + \ln x) \) is a primitive of \( g(x) = \frac{1}{x(1 + \ln x)} \).

### Method 2: Integration

Now, let's verify this result by calculating the integral of \( g(x) \).

#### Step 1: Set up the integral
We aim to compute the following integral:
\[
\int \frac{1}{x(1 + \ln x)} \, dx
\]

#### Step 2: Use substitution
Let \( u = 1 + \ln x \), so that \( du = \frac{1}{x} dx \). This transforms the integral into:
\[
\int \frac{1}{u} \, du
\]
which is a standard integral:
\[
\int \frac{1}{u} \, du = \ln|u| + C
\]

#### Step 3: Substitute back for \( u \)
Substituting \( u = 1 + \ln x \), we get:
\[
\ln(1 + \ln x) + C
\]

Thus, the integral of \( g(x) = \frac{1}{x(1 + \ln x)} \) is \( \ln(1 + \ln x) + C \), which confirms that \( f(x) = \ln(1 + \ln x) \) is a primitive of \( g(x) \).

### Conclusion
We have proven in two ways (by differentiation and by integration) that \( f(x) = \ln(1 + \ln x) \) is a primitive of \( g(x) = \frac{1}{x(1 + \ln x)} \).

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Do you have any questions or would you like more details?

Here are five related questions:
1. How can the chain rule be applied in other types of logarithmic functions?
2. What are the key differences between differentiation and integration in proving primitives?
3. How can substitution simplify more complex integrals like this one?
4. Can this technique be generalized for other compositions of functions?
5. How does one handle integrals involving multiple logarithms or nested functions?

**Tip:** Always check your integration results by differentiating the primitive to ensure correctness.

Să se demonstreze în două moduri că funcția f : (1, +∞) → ℝ, f(x) = ln(1 + ln x) este o primitivă a funcției g : (1, +∞) → ℝ, g(x) = 1 / [x(1 + ln x)].

Explore the solution to proving that the function f(x) = ln(1 + ln x) is a primitive of g(x) = 1 / [x(1 + ln x)] using both differentiation and integration techniques. This problem demonstrates the application of the chain rule and substitution method in calculus, suitable for advanced high school or undergraduate students.

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