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.

Solve the integration of (4/x³ + 2/x) dx

Learn how to solve the integral ∫ (4/x³ + 2/x) dx with a detailed step-by-step explanation. This solution uses key calculus concepts such as the power rule and logarithmic functions. Suitable for students in Grades 11-12 and early college.

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