The given problem requires calculating the indefinite integral:

$\int (x^3 + x^2 + x - 2) \, dx.$

Step-by-step solution:

1. Integrate each term separately:

   • The indefinite integral of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.

2. Apply to each term:

   • $\int x^3 \, dx = \frac{x^4}{4}$,
   • $\int x^2 \, dx = \frac{x^3}{3}$,
   • $\int x \, dx = \frac{x^2}{2}$,
   • $\int -2 \, dx = -2x$.

3. Combine results: $\int (x^3 + x^2 + x - 2) \, dx = \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} - 2x + C.$

Final Answer:

$\boxed{\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} - 2x + C}.$

Would you like further clarification or a deeper exploration into definite integrals or related topics?

Here are five follow-up questions to deepen your understanding:

1. How would you approach solving a definite integral with limits for this function?
2. Can you verify the result by differentiating the obtained function?
3. How would the integral change if an additional constant were added to the function?
4. What are the practical applications of integrating polynomial functions like this one?
5. How does this method extend to integrating more complex functions, such as those involving trigonometric terms?

Tip: Always check your integration by