The given integral is:

$\int (14x^3 - 15x^2 + 14x - 3) \, dx$

To solve this, we integrate term by term:

1. The integral of $14x^3$ is $\frac{14x^4}{4} = 3.5x^4$.
2. The integral of $-15x^2$ is $-\frac{15x^3}{3} = -5x^3$.
3. The integral of $14x$ is $\frac{14x^2}{2} = 7x^2$.
4. The integral of $-3$ is $-3x$.

Combining all the terms, the integral is:

$3.5x^4 - 5x^3 + 7x^2 - 3x + C$

where $C$ is the constant of integration.

Let me know if you want further details or have any questions!

Here are 5 related questions to expand your understanding:

1. How do you verify your solution by differentiating it?
2. What are the rules for integrating polynomial functions?
3. How does the integration process change with definite integrals?
4. Can you integrate a similar expression involving trigonometric functions?
5. How do you apply these principles to real-world problems?

Tip: Always double-check your solution by differentiating it to ensure it matches the original integrand.