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?

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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!