We are tasked with solving the integral:

$\int \left(\frac{4}{x^3} + \frac{2}{x}\right) dx$

Step 1: Split the integral

The integral can be split into two simpler parts:

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

Step 2: Solve each part separately

Part 1: Solve $\int \frac{4}{x^3} dx$

We rewrite $\frac{4}{x^3}$ as $4x^{-3}$, then apply the power rule for integration:

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

Part 2: Solve $\int \frac{2}{x} dx$

The integral of $\frac{2}{x}$ is a standard logarithmic integral:

$\int \frac{2}{x} dx = 2 \ln|x|$

Step 3: Combine the results

Now, combining both parts, we get:

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

where $C$ is the constant of integration.

Thus, the solution to the integral is:

$-\frac{2}{x^2} + 2 \ln|x| + C$

Would you like more details, or do you have any questions?

Here are some related questions:

1. How does the power rule for integration work for negative exponents?
2. Why is the logarithmic function used when integrating $\frac{1}{x}$?
3. How can integrals with different powers of $x$ be solved in more complex cases?
4. What is the importance of adding the constant $C$ in indefinite integrals?
5. Can this method be applied to integrals of rational functions?

Tip: Always check if the integrand can be simplified before applying integration techniques.