To solve the indefinite integral $\int (2x - 3 + x^2) \, dx$:

Step-by-step solution:

1. Break the integral into parts: $\int (2x - 3 + x^2) \, dx = \int 2x \, dx - \int 3 \, dx + \int x^2 \, dx$

2. Compute each term:

   • For $\int 2x \, dx$: Use the power rule, $\int x^n dx = \frac{x^{n+1}}{n+1}$: $\int 2x \, dx = x^2$
   • For $\int 3 \, dx$: The integral of a constant is the constant multiplied by $x$: $\int 3 \, dx = 3x$
   • For $\int x^2 \, dx$: Again, use the power rule: $\int x^2 \, dx = \frac{x^3}{3}$

3. Combine the results: $\int (2x - 3 + x^2) \, dx = x^2 - 3x + \frac{x^3}{3} + C$ where $C$ is the constant of integration.

Compare with options:

The correct answer matches option (d): $x^2 - 3x + \frac{1}{3}x^3 + C, \, C \in \mathbb{R}.$

Do you have any questions about this process or need further details? Here are some follow-up questions to expand your understanding:

1. What is the general rule for integrating a polynomial function?
2. How do constants affect the integration of a function?
3. Can you explain why the constant $C$ is always added in indefinite integrals?
4. How would the integral change if a different constant term was added to the original function?
5. What is the difference between definite and indefinite integrals?

Tip: Always simplify the expression inside the integral before starting to integrate!