https://file.aiquickdraw.com/mathbot/file/1725787120126mlaw6d5p.jpg

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.

Beräkna alla primitiva funktioner F till cos(4x) + sin(x).

Explore the step-by-step process of integrating the function cos(4x) + sin(x), including detailed solutions and related calculus concepts. Suitable for university or advanced pre-college calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation