The problem asks for the integral of the polynomial function $4x^3 - 6x^2 + x - 3$.

To solve the integral $\int (4x^3 - 6x^2 + x - 3) \, dx$, we integrate each term separately:

$\int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4$

$\int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} = 2x^3$

$\int x \, dx = \frac{x^2}{2}$

$\int (-3) \, dx = -3x$

Adding these results together, the integral is:

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

where $C$ is the constant of integration.

Would you like any further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine the constant of integration when evaluating indefinite integrals?
2. How does the power rule for integration differ from the power rule for differentiation?
3. What happens if one of the terms in the integrand is a fractional power or a root?
4. How would you integrate a polynomial function if it included trigonometric terms?
5. What are some real-world applications of integrating polynomial functions?

Tip: When integrating, always add a constant of integration ($C$) for indefinite integrals to account for the family of functions that differ by a constant.