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The integral you provided is:

\[
\int \frac{x^2}{x - 4} \, dx
\]

To solve this integral, we can use **polynomial division** because the degree of the numerator \(x^2\) is higher than that of the denominator \(x - 4\).

### Step 1: Perform Polynomial Division
We divide \(x^2\) by \(x - 4\).

1. Divide the leading term: \(\frac{x^2}{x} = x\).
2. Multiply \(x\) by \((x - 4)\), which gives \(x(x - 4) = x^2 - 4x\).
3. Subtract: \(x^2 - (x^2 - 4x) = 4x\).

Now, divide \(4x\) by \(x - 4\):

1. \(\frac{4x}{x} = 4\).
2. Multiply \(4\) by \((x - 4)\), which gives \(4(x - 4) = 4x - 16\).
3. Subtract: \(4x - (4x - 16) = 16\).

So, we have:

\[
\frac{x^2}{x - 4} = x + 4 + \frac{16}{x - 4}
\]

### Step 2: Integrate Each Term
Now, we integrate the expression term by term:

\[
\int \left( x + 4 + \frac{16}{x - 4} \right) \, dx
\]

1. \(\int x \, dx = \frac{x^2}{2}\)
2. \(\int 4 \, dx = 4x\)
3. \(\int \frac{16}{x - 4} \, dx = 16 \ln |x - 4|\)

### Step 3: Combine Results
Thus, the final result is:

\[
\frac{x^2}{2} + 4x + 16 \ln |x - 4| + C
\]

where \(C\) is the constant of integration.

Would you like further details on any step or additional clarifications?

Here are 5 related questions to deepen understanding:
1. How does polynomial division help in integrating rational functions?
2. Why do we need to break down the fraction into simpler terms for integration?
3. What are some alternative methods for solving integrals of rational functions?
4. How does the natural logarithm arise in the integration of rational functions?
5. What are some cases where polynomial division would not work for rational integrals?

**Tip:** When performing polynomial division, always make sure to align the terms by powers of \(x\) to avoid mistakes.

Integrate \( \frac{x^2}{x - 4} \) with respect to \(x\).

Learn how to solve the integral \( \int \frac{x^2}{x - 4} \, dx \) using polynomial division. This detailed step-by-step solution breaks down polynomial division and demonstrates how to integrate each term, including a natural logarithmic result.

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