To solve the integral \(\int x \ln(1 + x) \, dx\), we can apply **integration by parts**. 

Recall the formula for integration by parts:

\[
\int u \, dv = uv - \int v \, du
\]

We will choose \( u = \ln(1 + x) \) and \( dv = x \, dx \).

### Step 1: Differentiate \( u \) and integrate \( dv \)
- \( u = \ln(1 + x) \), so \( du = \frac{1}{1 + x} \, dx \)
- \( dv = x \, dx \), so \( v = \frac{x^2}{2} \)

### Step 2: Apply the integration by parts formula
Now substitute into the integration by parts formula:

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

### Step 3: Simplify the remaining integral
The remaining integral is:

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

We can simplify \(\frac{x^2}{1 + x}\) as follows:

\[
\frac{x^2}{1 + x} = x - 1 + \frac{1}{1 + x}
\]

Thus, the integral becomes:

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

This integral can now be separated:

\[
\frac{1}{2} \int x \, dx - \frac{1}{2} \int 1 \, dx + \frac{1}{2} \int \frac{1}{1 + x} \, dx
\]

### Step 4: Solve each part
- \(\int x \, dx = \frac{x^2}{2}\)
- \(\int 1 \, dx = x\)
- \(\int \frac{1}{1 + x} \, dx = \ln(1 + x)\)

### Step 5: Combine the results
Now, substitute the integrals back:

\[
\frac{1}{2} \left( \frac{x^2}{2} - x + \ln(1 + x) \right)
\]

### Step 6: Final answer
Substitute this result into the original equation:

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

Simplifying:

\[
\int x \ln(1 + x) \, dx = \frac{x^2}{2} \ln(1 + x) - \frac{x^2}{4} + \frac{x}{2} - \frac{\ln(1 + x)}{2} + C
\]

This is the final result.

---

Would you like further clarification or details on any step?

Here are 5 related questions to extend this topic:
1. How does integration by parts work for more complex functions?
2. What is the intuition behind choosing \( u \) and \( dv \) in integration by parts?
3. How would you solve the integral \(\int \ln(1 + x) \, dx\)?
4. What is the method for simplifying integrals involving logarithmic functions?
5. Can substitution be combined with integration by parts in certain integrals?

**Tip:** When applying integration by parts, it's often helpful to choose \( u \) as the term that simplifies when differentiated.

Solve the integral ∫ x ln(1 + x) dx step-by-step using the integration by parts method. Learn how logarithmic functions and calculus techniques apply in this advanced college-level problem.

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