To solve this integral, we are tasked with finding the antiderivative of the expression:

$\int \frac{5x^2 - 2}{x^2} dx$

Step-by-step solution:

1. Simplify the integrand: Split the fraction: $\frac{5x^2 - 2}{x^2} = \frac{5x^2}{x^2} - \frac{2}{x^2} = 5 - \frac{2}{x^2}$

   So, the integral becomes: $\int \left( 5 - \frac{2}{x^2} \right) dx$

2. Separate the integral: $\int \left( 5 - \frac{2}{x^2} \right) dx = \int 5 \, dx - \int \frac{2}{x^2} \, dx$

3. Integrate each term:

   • For $\int 5 \, dx$, the result is $5x$.
   • For $\int \frac{-2}{x^2} \, dx$, rewrite $\frac{1}{x^2}$ as $x^{-2}$ and integrate: $\int -2x^{-2} \, dx = -2 \cdot \frac{x^{-1}}{-1} = \frac{2}{x}.$

4. Combine the results: The final answer is: $5x + \frac{2}{x} + C$

Final Answer:

$\int \frac{5x^2 - 2}{x^2} \, dx = 5x + \frac{2}{x} + C$

Let me know if you have any questions or would like further clarification!

Here are 5 follow-up questions to deepen understanding:

1. How do we handle integrals involving negative exponents like $x^{-2}$?
2. What is the role of the constant $C$ in indefinite integrals?
3. Can you explain why splitting the fraction simplifies the integration process?
4. What is the difference between definite and indefinite integrals?
5. How would the solution change if the integral included limits?

Tip: Always simplify the integrand as much as possible before attempting to integrate—it often makes the calculations easier!