We are given the equation:

$\int f(x) dx = \frac{1}{4}x^4 + 2x^3 + \frac{1}{2}x^2 + 3x + C$

and need to find $f(x)$, where $C$ is the constant of integration.

To find $f(x)$, we differentiate both sides of the equation with respect to $x$. Using the fact that the derivative of an integral gives us back the original function $f(x)$, and applying standard differentiation rules to the right-hand side:

$f(x) = \frac{d}{dx} \left( \frac{1}{4}x^4 + 2x^3 + \frac{1}{2}x^2 + 3x + C \right)$

The derivative is computed as follows:

• The derivative of $\frac{1}{4}x^4$ is $x^3$,
• The derivative of $2x^3$ is $6x^2$,
• The derivative of $\frac{1}{2}x^2$ is $x$,
• The derivative of $3x$ is $3$,
• The derivative of $C$ (a constant) is 0.

Thus, we have:

$f(x) = x^3 + 6x^2 + x + 3$

Looking at the provided choices, the correct answer is:

$\boxed{③ \, x^3 + 6x^2 + x + 3}$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What are the rules for differentiating polynomial functions?
2. How do constants affect the process of integration and differentiation?
3. What is the difference between indefinite and definite integrals?
4. How can you verify the correctness of a function through differentiation?
5. Why do we use the constant $C$ in indefinite integrals?

Tip: Always differentiate each term carefully to avoid missing powers of $x$ or coefficients!