To solve the integral

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

we can rewrite and separate the terms for easier integration:

Step 1: Split the terms

$\int \frac{2\sqrt{x} - 3}{x^2} \, dx = \int \frac{2\sqrt{x}}{x^2} \, dx - \int \frac{3}{x^2} \, dx.$

Step 2: Simplify each term

1. The first term: $\frac{\sqrt{x}}{x^2} = \frac{x^{1/2}}{x^2} = x^{1/2 - 2} = x^{-3/2}.$

2. The second term remains: $\frac{3}{x^2} = 3x^{-2}.$

Thus, the integral becomes: $\int x^{-3/2} \cdot 2 \, dx - \int 3x^{-2} \, dx.$

Step 3: Integrate each term

1. For $\int 2x^{-3/2} \, dx$: $\int x^{-3/2} \, dx = \frac{x^{-3/2+1}}{-3/2+1} = \frac{x^{-1/2}}{-1/2} = -2x^{-1/2}.$ Multiply by 2: $-4x^{-1/2}.$

2. For $\int 3x^{-2} \, dx$: $\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -x^{-1}.$ Multiply by 3: $-3x^{-1}.$

Step 4: Combine the results

$\int \frac{2\sqrt{x} - 3}{x^2} \, dx = -4x^{-1/2} - 3x^{-1} + C,$ where $C$ is the constant of integration.

Step 5: Simplify the answer

Using positive exponents and radicals: $\boxed{-\frac{4}{\sqrt{x}} - \frac{3}{x} + C}.$

Would you like a detailed explanation for any part of this? Here are some related questions to extend your understanding:

1. How does splitting the terms simplify the integral?
2. Why do we adjust the exponents when integrating $x^n$?
3. What is the significance of the constant $C$ in indefinite integrals?
4. Can this integral be solved using substitution methods? If yes, how?
5. How can this integral be verified by differentiation?

Tip: Always rewrite complex fractions to separate terms for easier integration!