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The integral you provided is:

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

To solve this integral, we can use a technique called **polynomial division** (if applicable) or **substitution** to simplify the integrand.

### Solution Steps

1. **Rewrite the Integrand**:
   Since \( x^2 \) is of higher degree than \( 2 + x \), let's use polynomial division to rewrite the fraction.

   Dividing \( x^2 \) by \( 2 + x \), we get:
   \[
   x^2 = x \cdot (2 + x) - 2x
   \]
   So,
   \[
   \frac{x^2}{2 + x} = x - \frac{2x}{2 + x}
   \]
   We now rewrite the integral as:
   \[
   \int \frac{x^2}{2 + x} \, dx = \int \left( x - \frac{2x}{2 + x} \right) dx
   \]

2. **Separate the Integral**:
   Now, split the integral into two parts:
   \[
   \int x \, dx - \int \frac{2x}{2 + x} \, dx
   \]

3. **Evaluate Each Integral**:
   - For the first integral, \(\int x \, dx = \frac{x^2}{2}\).
   - For the second integral, we use **substitution**:
     Let \( u = 2 + x \), so \( du = dx \) and \( x = u - 2 \).
     
     Substitute into the integral:
     \[
     \int \frac{2x}{2 + x} \, dx = \int \frac{2(u - 2)}{u} \, du = \int 2 \, du - \int \frac{4}{u} \, du
     \]
     This simplifies to:
     \[
     2u - 4 \ln |u|
     \]

4. **Combine Results**:
   Now substitute back for \( u = 2 + x \) in the expression and combine terms.

Let me know if you would like the detailed steps for each part, or if you have any questions.

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### Related Questions
1. How do you handle integrals involving higher degree polynomials in the numerator?
2. What other substitution methods could simplify integrals like this?
3. Can we use partial fractions on similar integrals?
4. How does polynomial division work in integrals?
5. What is the role of logarithmic functions in integrals involving polynomial denominators?

### Tip:
For integrals with polynomial numerators and denominators, always check if polynomial division simplifies the expression before attempting substitution.

Calculate the integral \( \int \frac{x^2}{2 + x} \, dx \).

This solution demonstrates the step-by-step calculation of the integral \( \int \frac{x^2}{2 + x} \, dx \) using polynomial division and substitution techniques. Perfect for college-level calculus students, this problem covers integration strategies for rational functions with polynomials.

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