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To solve the integral:

\[
\int \frac{dx}{(x+1) \ln(x+1)}, \quad x > -1
\]

we'll proceed with substitution.

1. **Substitute** \( u = \ln(x+1) \). Then \( du = \frac{1}{x+1} \, dx \).
   
   This transforms the integral into:
   
   \[
   \int \frac{1}{u} \, du
   \]

2. **Integrate** \( \frac{1}{u} \):
   
   \[
   \int \frac{1}{u} \, du = \ln |u| + C
   \]

3. **Substitute back** \( u = \ln(x+1) \):
   
   \[
   \ln |\ln(x+1)| + C
   \]

Thus, the correct answer is:

\[
\text{B. } \ln |\ln(x+1)| + C, \quad C \in \mathbb{R}
\]

Would you like further details on any part of the solution?

Here are 5 related questions to consider:
1. How would you integrate if the integral had a slightly different logarithmic function in the denominator?
2. What would happen if the limits of integration were specified for this integral?
3. Can you generalize this method to other integrals involving nested logarithmic functions?
4. How would this integral change if \(x\) were replaced with another variable with a different function in the denominator?
5. How does this method apply to integrals with trigonometric substitutions?

**Tip:** When encountering integrals with nested functions like logarithms, consider substitution to simplify the expression first.

Compute the following indefinite integral ∫ dx / ((x + 1) ln(x + 1)), x > -1.

Learn how to compute the indefinite integral ∫ dx / ((x + 1) ln(x + 1)) using substitution and logarithmic integration. This solution involves key calculus concepts and is suitable for college-level students.

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