In the image you provided, the problem asks to find the integral of the expression:

$\int \frac{x^3 + 2}{x^2 + 4} \, dx$

The solution proceeds with a series of steps to simplify the expression and solve the integral. Let's break it down:

1. Splitting the Expression: The integrand $\frac{x^3 + 2}{x^2 + 4}$ is broken down into two parts by separating the numerator:

$\int \frac{x^3 + 2}{x^2 + 4} \, dx = \int \frac{x^3 + 4x - 4x + 2}{x^2 + 4} \, dx$

This is done to make the integral easier to handle by splitting it into two separate integrals:

$\int \frac{x^3 + 4x}{x^2 + 4} \, dx - \int \frac{4x - 2}{x^2 + 4} \, dx$

2. Simplifying the First Integral: Now, the first part $\int \frac{x^3 + 4x}{x^2 + 4} \, dx$ is simplified to:

$\int x \, dx + \int \frac{1}{2} \, \frac{dx}{(x/2)^2 + 1}$

This represents a simpler form where we can apply basic integration rules.

3. Second Integral: The second integral $- \int \frac{x}{x^2 + 4} \, dx$ is a standard form. Using the substitution $u = x/2$, this turns into an inverse tangent function:

$- \int \frac{dx}{x^2 + 4} = - \frac{1}{2} \, \text{arctan}\left(\frac{x}{2}\right)$

4. Final Result: After performing all integrations and collecting the results, the answer to the integral is:

$\int \frac{x^3 + 2}{x^2 + 4} \, dx = \frac{x^2}{2} + \text{arctan}\left(\frac{x}{2}\right) - 2 \ln |x^2 + 4| + c$

This is the final solution for the integral.

Would you like any further explanation of the steps, or do you have any specific questions about them?

Here are five related questions that might help deepen your understanding:

1. How do we handle integrals of rational functions with quadratic denominators?
2. Can you explain the process of splitting a rational function into simpler integrals?
3. How do you recognize the need for substitution when solving integrals?
4. What is the significance of the arctangent function in integration?
5. How do you handle the logarithmic term that arises during integration?

Tip: Always check if the integrand can be split into simpler terms or if a substitution can simplify the integral. This often makes finding the solution much easier.