If `intf(x)dx=F(x),` then `int(f(x))/(F'(x))dx=`

Options

A. `x+c`
B. `logf(x)+c`
C. `logF(x)+c`
D. `e^(f(x))+c`

Answer

To solve the integral \( \int \frac{f(x)}{F'(x)} \, dx \) where \( F(x) = \int f(x) \, dx \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- We have \( F(x) = \int f(x) \, dx \).
- By the Fundamental Theorem of Calculus, we know that \( F'(x) = f(x) \).

2. **Substitute \( F'(x) \)**:
- Since \( F'(x) = f(x) \), we can rewrite the integral:
\[ \int \frac{f(x)}{F'(x)} \, dx = \int \frac{f(x)}{f(x)} \, dx \]

3. **Simplify the Integral**:
- The expression \( \frac{f(x)}{f(x)} \) simplifies to 1 (as long as \( f(x) \neq 0 \)):
\[ \int 1 \, dx \]

4. **Integrate**:
- The integral of 1 with respect to \( x \) is:
\[ x + C \]
where \( C \) is the constant of integration.

5. **Final Answer**:
- Therefore, we conclude that:
\[ \int \frac{f(x)}{F'(x)} \, dx = x + C \]