The problem asks to calculate all the antiderivatives (primitive functions) $F$ of $\cos(4x) + \sin(x)$.

To solve this, we need to integrate the function $f(x) = \cos(4x) + \sin(x)$.

Step-by-step solution:

1. Integral of $\cos(4x)$: $\int \cos(4x) \, dx = \frac{1}{4} \sin(4x)$ (Using the standard rule for integrating $\cos(ax)$: $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax)$)

2. Integral of $\sin(x)$: $\int \sin(x) \, dx = -\cos(x)$ (This is a standard integral for $\sin(x)$).

3. Combining the two integrals: $\int (\cos(4x) + \sin(x)) \, dx = \frac{1}{4} \sin(4x) - \cos(x) + C$ where $C$ is the constant of integration.

Final Answer:

$F(x) = \frac{1}{4} \sin(4x) - \cos(x) + C$

Let me know if you'd like further details or clarifications!

Related Questions:

1. What is the derivative of $\frac{1}{4} \sin(4x) - \cos(x)$?
2. How do you integrate $\cos(4x)$ with different limits?
3. Can you generalize the integral $\int \cos(ax) \, dx$?
4. How would the integration change if the function was $\cos(4x) - \sin(x)$?
5. How do we find definite integrals for $\cos(4x) + \sin(x)$ between specific bounds?

Tip:

Always remember to add the constant of integration $C$ when calculating indefinite integrals.