To solve for the constant \( C \) in the equation 

\[
\int f'(x) \, dx = f(x) + C,
\]

we need to understand the role of \( C \) in the equation. The constant \( C \) represents the arbitrary constant of integration that arises when we take the indefinite integral of a function.

### Steps to Solve for \( C \):

1. **Consider the general solution**:
   The equation states that the indefinite integral of \( f'(x) \) with respect to \( x \) is \( f(x) \) plus a constant \( C \).

   \[
   \int f'(x) \, dx = f(x) + C
   \]

2. **Apply the Fundamental Theorem of Calculus**:
   According to the Fundamental Theorem of Calculus, the integral of a derivative function \( f'(x) \) is the original function \( f(x) \), plus an arbitrary constant \( C \).

   \[
   \int f'(x) \, dx = f(x) + C
   \]

3. **Determine \( C \)**:
   To solve for \( C \), you need an initial condition, often provided as \( f(a) = b \) for some specific point \( a \) where \( f(x) \) is known.

   Let's say you know \( f(x) \) at some specific point \( x = x_0 \), i.e., \( f(x_0) = y_0 \). Substitute this into the equation:

   \[
   f(x_0) + C = y_0
   \]

   This gives:

   \[
   C = y_0 - f(x_0)
   \]

Without an initial condition, \( C \) remains an arbitrary constant.

### Conclusion:
The constant \( C \) is determined by the specific value of \( f(x) \) at a given point, which is an initial condition or boundary condition.

Would you like to go into more details on any part of this, or do you have any other questions?

### Relative Questions:
1. How does the Fundamental Theorem of Calculus apply to solving integrals?
2. What are initial conditions, and why are they necessary to determine \( C \)?
3. Can the constant \( C \) be negative or zero?
4. How would you solve for \( C \) if multiple initial conditions are given?
5. What happens if \( f(x) \) is a constant function?
6. How does the integration process change if \( f'(x) \) is a trigonometric function?
7. What are some common mistakes when solving for \( C \)?
8. How does definite integration differ from indefinite integration regarding constants?

### Tip:
Always check if initial conditions are provided before solving for the constant \( C \), as they are crucial for determining its exact value.

Learn how to solve for the constant C in the integral equation ∫ f'(x) dx = f(x) + C with a detailed step-by-step solution. This problem covers calculus concepts such as the Fundamental Theorem of Calculus and the role of constants of integration. Suitable for advanced high school students and beyond.

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