We want to evaluate the integral:

\[
\int \frac{x^2 - x + 6}{x^3 + 3x} \, dx
\]

### Step 1: Simplify the integrand

The denominator can be factored as:

\[
x^3 + 3x = x(x^2 + 3)
\]

So, the integral becomes:

\[
\int \frac{x^2 - x + 6}{x(x^2 + 3)} \, dx
\]

### Step 2: Split the fraction

We can split the fraction into simpler parts by dividing each term in the numerator by \(x(x^2 + 3)\):

\[
\frac{x^2 - x + 6}{x(x^2 + 3)} = \frac{x^2}{x(x^2 + 3)} - \frac{x}{x(x^2 + 3)} + \frac{6}{x(x^2 + 3)}
\]

This simplifies to:

\[
\frac{x^2}{x(x^2 + 3)} = \frac{1}{x^2 + 3}, \quad \frac{x}{x(x^2 + 3)} = \frac{1}{x^2 + 3}, \quad \frac{6}{x(x^2 + 3)}
\]

Thus, the integrand is now:

\[
\int \left( \frac{1}{x^2 + 3} + \frac{5}{x(x^2 + 3)} \right) dx
\]

### Step 3: Solve the integral

1. The first part \( \int \frac{1}{x^2 + 3} \, dx \) is a standard integral:

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

2. For the second part \( \int \frac{5}{x(x^2 + 3)} \, dx \), we can solve this using a substitution. Let \( u = x^2 + 3 \), then \( du = 2x \, dx \), and this becomes a standard logarithmic integral. After some algebraic steps, we obtain:

\[
\int \frac{5}{x(x^2 + 3)} \, dx = \frac{5}{6} \ln|x^2 + 3| + C
\]

### Step 4: Combine the results

So, the final answer for the integral is:

\[
\int \frac{x^2 - x + 6}{x^3 + 3x} \, dx = \frac{1}{\sqrt{3}} \arctan\left(\frac{x}{\sqrt{3}}\right) + \frac{5}{6} \ln|x^2 + 3| + C
\]

Would you like further clarification or a deeper breakdown of any steps?

---

Here are five related questions to expand your understanding:

1. How would you approach solving \( \int \frac{6}{x(x^2 + 3)} \, dx \) separately?
2. Can you explain the technique of using substitution to simplify integrals?
3. What is the derivative of \( \arctan\left(\frac{x}{\sqrt{3}}\right) \)?
4. How does partial fraction decomposition apply to integrals of rational functions?
5. What other methods can you use to solve complex rational integrals?

**Tip:** Always check if the denominator can be factored or simplified before attempting an integral—it can make the process much easier!

Learn how to solve the integral of (x^2 - x + 6) / (x^3 + 3x) with a detailed step-by-step solution. This advanced calculus problem explores algebraic fractions, substitution, logarithmic, and trigonometric functions. Suitable for students in Grades 11-12.

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