To evaluate the integral of the function \(\frac{x^3 - 2x + 1}{x^2 + 4}\), we can use polynomial division and partial fraction decomposition to simplify the expression before integrating. Here’s the step-by-step approach:

### Step 1: Simplify the Function Using Polynomial Division

Given the integral:

\[
\int \frac{x^3 - 2x + 1}{x^2 + 4} \, dx
\]

we perform polynomial division on \( \frac{x^3 - 2x + 1}{x^2 + 4} \).

### Step 2: Perform Polynomial Division

Divide \( x^3 - 2x + 1 \) by \( x^2 + 4 \):

1. Divide the leading term \(x^3\) by \(x^2\) to get \(x\).
2. Multiply \(x\) by \(x^2 + 4\) to get \(x^3 + 4x\).
3. Subtract this from the original polynomial: 

   \[
   (x^3 - 2x + 1) - (x^3 + 4x) = -6x + 1.
   \]

Now, the expression simplifies to:

\[
\frac{x^3 - 2x + 1}{x^2 + 4} = x + \frac{-6x + 1}{x^2 + 4}.
\]

### Step 3: Rewrite the Integral

The integral can now be rewritten as:

\[
\int \left(x + \frac{-6x + 1}{x^2 + 4}\right) dx = \int x \, dx + \int \frac{-6x + 1}{x^2 + 4} \, dx.
\]

### Step 4: Integrate Each Term Separately

1. **Integrate \(\int x \, dx\):**

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

2. **Integrate \(\int \frac{-6x + 1}{x^2 + 4} \, dx\):**

   We can split the fraction:

   \[
   \frac{-6x + 1}{x^2 + 4} = -6 \cdot \frac{x}{x^2 + 4} + \frac{1}{x^2 + 4}.
   \]

   Now, integrate these separately:

   - For \(\int \frac{x}{x^2 + 4} \, dx\), use the substitution \( u = x^2 + 4 \), leading to \( du = 2x \, dx \).
   - For \(\int \frac{1}{x^2 + 4} \, dx\), recognize it as a standard arctangent form.

Let's compute these integrals step by step.The integral of the given function \(\frac{x^3 - 2x + 1}{x^2 + 4}\) is:

\[
\int \frac{x^3 - 2x + 1}{x^2 + 4} \, dx = \frac{x^2}{2} - 3\ln(x^2 + 4) + \frac{\arctan\left(\frac{x}{2}\right)}{2} + C
\]

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

### Summary of Steps:

1. **Polynomial Division:** Simplified the original function using division.
2. **Integration:** Integrated each part separately, utilizing logarithmic and arctangent functions.

Would you like further details on any part of the solution, or do you have any questions? 

Here are some related questions you might consider:

1. How does substitution work in integration?
2. What is the role of partial fractions in integration?
3. Can we apply similar steps to integrate different rational functions?
4. What are the properties of the arctangent function?
5. How does polynomial division help simplify complex integrals?

**Tip:** Understanding polynomial division helps in breaking down complex rational functions, making integration manageable!

Learn how to integrate the function (x^3 - 2x + 1) / (x^2 + 4) step-by-step with polynomial division and partial fraction decomposition. This problem covers advanced integration techniques, including polynomial division and partial fraction decomposition. Suitable for advanced high school or college-level students.

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