We are tasked with evaluating the integral:

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

Step 1: Simplify the integrand

We simplify the expression $\frac{x^2 + 3x - 2}{x}$ by dividing each term in the numerator by $x$:

$\frac{x^2 + 3x - 2}{x} = \frac{x^2}{x} + \frac{3x}{x} - \frac{2}{x}.$

Simplify each term:

$\frac{x^2}{x} = x, \quad \frac{3x}{x} = 3, \quad \frac{-2}{x} = -\frac{2}{x}.$

Thus, the integral becomes:

$\int \frac{x^2 + 3x - 2}{x} \, dx = \int \left( x + 3 - \frac{2}{x} \right) \, dx.$

Step 2: Integrate term by term

We now integrate each term separately:

1. $\int x \, dx = \frac{x^2}{2}$,
2. $\int 3 \, dx = 3x$,
3. $\int -\frac{2}{x} \, dx = -2 \ln |x|$.

Step 3: Combine results

Combining the results of the individual integrals, we get:

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

where $C$ is the constant of integration.

Final Answer:

$\int \frac{x^2 + 3x - 2}{x} \, dx = \frac{x^2}{2} + 3x - 2 \ln |x| + C.$

Would you like a breakdown of any step in detail or an explanation of a particular concept? Let me know!

Related Questions:

1. How do you integrate a polynomial divided by a monomial step by step?
2. What is the general formula for integrating $\frac{1}{x}$?
3. How can partial fraction decomposition be applied to more complex rational functions?
4. Can you explain how to check the derivative of the integral to verify the result?
5. What happens if the integral involves a more complicated denominator?

Tip: Always simplify the integrand first when dealing with polynomials or rational functions. This makes the integration process easier!