To solve the given definite integral:

$\int_{3}^{1} (3x^2 - x + 1) \, dx$

First, we find the antiderivative (indefinite integral) of the function $3x^2 - x + 1$.

1. The antiderivative of $3x^2$ is $x^3$.
2. The antiderivative of $-x$ is $-\frac{x^2}{2}$.
3. The antiderivative of $1$ is $x$.

Combining these, the antiderivative $F(x)$ of $3x^2 - x + 1$ is:

$F(x) = x^3 - \frac{x^2}{2} + x$

Now, we evaluate this antiderivative at the upper and lower limits of the integral:

$\int_{3}^{1} (3x^2 - x + 1) \, dx = F(1) - F(3)$

Calculating $F(1)$:

$F(1) = 1^3 - \frac{1^2}{2} + 1 = 1 - 0.5 + 1 = 1.5$

Calculating $F(3)$:

$F(3) = 3^3 - \frac{3^2}{2} + 3 = 27 - 4.5 + 3 = 25.5$

Subtract $F(3)$ from $F(1)$:

$F(1) - F(3) = 1.5 - 25.5 = -24$

Therefore, the value of the definite integral is:

$\boxed{-24}$

Would you like any additional details or have any questions?

Here are 8 relative questions to expand on this information:

1. What is the process for finding the antiderivative of a polynomial function?
2. How do you evaluate a definite integral using the Fundamental Theorem of Calculus?
3. Can you explain how to interpret the limits of integration in a definite integral?
4. What are some common mistakes to avoid when computing definite integrals?
5. How can definite integrals be applied in real-world scenarios?
6. What is the significance of the area under a curve in integral calculus?
7. How do you handle integrals of more complex functions that are not simple polynomials?
8. Can you explain the concept of an improper integral and how it differs from a regular definite integral?

Tip: When solving definite integrals, always double-check the limits of integration and ensure you correctly substitute them into the antiderivative function.