integrate (x ^ 3 - 2x + 1)/(x ^ 2 + 4) dx

To solve the integral \(\int \frac{x^3 - 2x + 1}{x^2 + 4} \, dx\), we can proceed by dividing the integrand into simpler parts. Here's the step-by-step process:

### Step 1: Simplify the integrand by division
Divide \(x^3 - 2x + 1\) by \(x^2 + 4\) using polynomial long division.

1. **Divide**: \(x^3\) by \(x^2\) gives \(x\).
2. **Multiply**: \(x(x^2 + 4) = x^3 + 4x\).
3. **Subtract**: \((x^3 - 2x + 1) - (x^3 + 4x) = -6x + 1\).

So, the result of the division is:
\[
\frac{x^3 - 2x + 1}{x^2 + 4} = x - \frac{6x - 1}{x^2 + 4}.
\]
Thus, the integral becomes:
\[
\int \frac{x^3 - 2x + 1}{x^2 + 4} \, dx = \int \left(x - \frac{6x - 1}{x^2 + 4}\right) dx.
\]

### Step 2: Integrate each term separately
Now, let's integrate each term separately:

1. **First term**:
   \[
   \int x \, dx = \frac{x^2}{2}.
   \]

2. **Second term**: 
   \[
   \int \frac{6x - 1}{x^2 + 4} \, dx.
   \]
   Let's break this into two parts:
   - \( \int \frac{6x}{x^2 + 4} \, dx \).
   - \( \int \frac{-1}{x^2 + 4} \, dx \).

#### Part 1: \( \int \frac{6x}{x^2 + 4} \, dx \)
Let \( u = x^2 + 4 \), then \( du = 2x \, dx \).
So,
\[
\int \frac{6x}{x^2 + 4} \, dx = 3 \int \frac{2x}{x^2 + 4} \, dx = 3 \int \frac{du}{u} = 3 \ln|u| = 3 \ln(x^2 + 4).
\]

#### Part 2: \( \int \frac{-1}{x^2 + 4} \, dx \)
This can be recognized as a standard arctangent integral:
\[
\int \frac{-1}{x^2 + 4} \, dx = -\frac{1}{2} \arctan\left(\frac{x}{2}\right).
\]

### Step 3: Combine all parts
Now, combining all the parts together:
\[
\int \frac{x^3 - 2x + 1}{x^2 + 4} \, dx = \frac{x^2}{2} + 3 \ln(x^2 + 4) - \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C,
\]
where \(C\) is the constant of integration.

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

Would you like further details on any of the steps? Here are some related questions you might find interesting:

1. How do you perform polynomial long division in integrals?
2. What is the standard approach to integrating rational functions?
3. How does the substitution method work in integration?
4. Can you explain more about the arctangent integral?
5. How can partial fractions be used in integrals similar to this one?

**Tip:** Recognizing standard forms like \(\int \frac{1}{x^2 + a^2} \, dx\) can significantly simplify complex integrals.

Explore how to integrate (x^3 - 2x + 1)/(x^2 + 4) step-by-step. Learn about polynomial long division in integration, handling rational functions, and the arctangent integral. Suitable for advanced college-level students.

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