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To find the position function \( s(t) \) from the given velocity function \( v(t) = 7 \cos(t) + 2 \sin(t) \) and initial condition \( s(0) = 4 \), we need to integrate the velocity function with respect to \( t \).

### Solution:
1. **Integrate \( v(t) \) to find \( s(t) \):**

   \[
   s(t) = \int v(t) \, dt = \int (7 \cos(t) + 2 \sin(t)) \, dt
   \]

2. **Compute the integral:**

   - The integral of \( 7 \cos(t) \) is \( 7 \sin(t) \).
   - The integral of \( 2 \sin(t) \) is \( -2 \cos(t) \).

   So, we get:

   \[
   s(t) = 7 \sin(t) - 2 \cos(t) + C
   \]

3. **Determine the constant \( C \) using the initial condition \( s(0) = 4 \):**

   Substitute \( t = 0 \) and \( s(0) = 4 \):

   \[
   s(0) = 7 \sin(0) - 2 \cos(0) + C = 4
   \]

   Since \( \sin(0) = 0 \) and \( \cos(0) = 1 \), we have:

   \[
   0 - 2 + C = 4
   \]

   \[
   C = 6
   \]

4. **Write the final position function:**

   \[
   s(t) = 7 \sin(t) - 2 \cos(t) + 6
   \]

### Answer:
\[
s(t) = 7 \sin(t) - 2 \cos(t) + 6
\]

Would you like a more detailed explanation of any step?

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Here are some related questions that might help expand your understanding:

1. How would you find the position function if the initial condition was different, say \( s(0) = 10 \)?
2. What is the physical meaning of the constant \( C \) in this context?
3. How does the integration of velocity relate to displacement and position in motion problems?
4. How would the solution change if \( v(t) \) included a \( t \)-dependent term like \( t \cos(t) \)?
5. What are some other real-world applications of integrating a velocity function?

**Tip:** Always remember that integration of a velocity function yields the position function, with an arbitrary constant that’s determined by initial conditions.

A particle is moving with the given data. Find the position of the particle, s(t).

v(t) = 7 cos(t) + 2 sin(t),  s(0) = 4

Learn how to find the position of a particle given its velocity function v(t) = 7 cos(t) + 2 sin(t) and initial condition s(0) = 4. This calculus problem demonstrates the integration of a velocity function to determine position, using the Fundamental Theorem of Calculus.

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