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Let's solve the given integral step by step. The integral is:

\[
\int \frac{dx}{(x + 1) \sqrt{x^2 + 2x + 1}}
\]

### Step 1: Simplify the square root expression

Notice that the expression under the square root can be simplified:

\[
x^2 + 2x + 1 = (x + 1)^2
\]

So, the integral becomes:

\[
\int \frac{dx}{(x + 1) \sqrt{(x + 1)^2}}
\]

### Step 2: Simplify the square root

Since \(\sqrt{(x + 1)^2} = |x + 1|\), the integral now becomes:

\[
\int \frac{dx}{(x + 1) |x + 1|}
\]

### Step 3: Consider the absolute value

There are two cases to consider for the absolute value:

1. **Case 1:** \(x + 1 \geq 0\) (i.e., \(x \geq -1\))
   
   In this case, \(|x + 1| = x + 1\). The integral simplifies to:

   \[
   \int \frac{dx}{(x + 1)(x + 1)} = \int \frac{dx}{(x + 1)^2}
   \]

   The integral of \(\frac{1}{(x + 1)^2}\) is:

   \[
   \int \frac{dx}{(x + 1)^2} = -\frac{1}{x + 1} + C
   \]

2. **Case 2:** \(x + 1 < 0\) (i.e., \(x < -1\))
   
   In this case, \(|x + 1| = -(x + 1)\). The integral simplifies to:

   \[
   \int \frac{dx}{(x + 1)(-(x + 1))} = \int -\frac{dx}{(x + 1)^2}
   \]

   The integral of \(-\frac{1}{(x + 1)^2}\) is:

   \[
   \int -\frac{dx}{(x + 1)^2} = \frac{1}{x + 1} + C
   \]

### Final Answer

The final result for the integral is:

\[
\int \frac{dx}{(x + 1) \sqrt{x^2 + 2x + 1}} = 
\begin{cases}
-\frac{1}{x + 1} + C & \text{if } x \geq -1 \\
\frac{1}{x + 1} + C & \text{if } x < -1
\end{cases}
\]

Would you like any further details or explanations?

Here are some related questions that might interest you:

1. What happens when the integrand is a more complex function?
2. How do you handle absolute values in integrals generally?
3. Can we solve the integral using a substitution method?
4. What if the quadratic inside the square root wasn’t a perfect square?
5. How does the sign of the function affect the integral result?
6. How do definite integrals work with these kinds of absolute value cases?
7. What are some examples of integrals involving more complex absolute values?
8. Can this integral be approached using a different method, such as partial fractions?

**Tip:** When solving integrals involving square roots, always check if the expression inside the square root can be factored or simplified—it can make the problem much easier!

Explore a detailed step-by-step solution to the integral ∫ dx / (x + 1) sqrt(x^2 + 2x + 1). Learn how to handle absolute values and integrate complex expressions effectively. Suitable for college-level calculus students.

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