To solve the integral:

$\int 4x^2 \left( \frac{2}{x^2} + \frac{3}{5x^4} \right) \, dx$

we'll proceed by simplifying the integrand first.

Step 1: Distribute $4x^2$ to each term inside the parentheses

$4x^2 \cdot \frac{2}{x^2} + 4x^2 \cdot \frac{3}{5x^4}$

This gives: $= \frac{8x^2}{x^2} + \frac{12x^2}{5x^4}$

Step 2: Simplify each term

1. For $\frac{8x^2}{x^2}$: $\frac{8x^2}{x^2} = 8$

2. For $\frac{12x^2}{5x^4}$: $\frac{12x^2}{5x^4} = \frac{12}{5x^2}$

So the integral simplifies to: $\int \left( 8 + \frac{12}{5x^2} \right) \, dx$

Step 3: Integrate term by term

1. For $\int 8 \, dx$: $\int 8 \, dx = 8x$

2. For $\int \frac{12}{5x^2} \, dx$: Rewrite $\frac{12}{5x^2}$ as $\frac{12}{5} x^{-2}$: $\int \frac{12}{5} x^{-2} \, dx = \frac{12}{5} \cdot \frac{-1}{x} = -\frac{12}{5x}$

Step 4: Combine results

So, the integral is: $8x - \frac{12}{5x} + C$

where $C$ is the constant of integration.

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Would you like more details on any steps, or have any questions?

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Here are some related questions you might find interesting:

1. How do we approach integrals involving multiple terms in the denominator?
2. What methods are used to simplify complex rational functions before integration?
3. How does the power rule apply in integration, especially with negative exponents?
4. How do we determine constants in indefinite integrals?
5. What are common pitfalls when distributing terms in integration problems?

Tip: When integrating terms with $x$ in both the numerator and denominator, simplify each term independently before attempting to integrate.