The integral of the function \( \frac{f(x)}{F(x)} \), where \( F(x) \) is an antiderivative of \( f(x) \), can be solved using substitution.

### Step-by-Step Solution:

1. **Recognize the relationship between \( f(x) \) and \( F(x) \):**
   Since \( F(x) \) is an antiderivative of \( f(x) \), we have:
   \[
   F'(x) = f(x)
   \]

2. **Substitute \( u = F(x) \):**
   Let \( u = F(x) \), then \( du = F'(x) dx = f(x) dx \).

3. **Rewrite the integral:**
   The integral becomes:
   \[
   \int \frac{f(x)}{F(x)} dx = \int \frac{1}{u} du
   \]

4. **Integrate \( \frac{1}{u} \) with respect to \( u \):**
   The integral of \( \frac{1}{u} \) is:
   \[
   \int \frac{1}{u} du = \ln |u| + C
   \]
   where \( C \) is the constant of integration.

5. **Substitute back \( u = F(x) \):**
   Replacing \( u \) with \( F(x) \), the final answer is:
   \[
   \int \frac{f(x)}{F(x)} dx = \ln |F(x)| + C
   \]

### Final Answer:
\[
\int \frac{f(x)}{F(x)} dx = \ln |F(x)| + C
\]

This is the general solution for the integral of \( \frac{f(x)}{F(x)} \).

Would you like to dive into more details, or have any questions? Here are some related problems to consider:

1. What is the integral of \( \frac{2x}{x^2 + 1} \)?
2. How would you evaluate \( \int \frac{1}{\sin x} \) using substitution?
3. Can you solve \( \int \frac{e^x}{e^x + 1} dx \)?
4. What is the integral of \( \frac{x^2}{\ln(x)} \)?
5. How do you approach \( \int \frac{\cos(x)}{\sin(x)} dx \)?
6. What is the integral of \( \frac{x^3}{x^4 + 1} \)?
7. How would you solve \( \int \frac{\tan(x)}{\sec(x)} dx \)?
8. Can you find the integral of \( \frac{2x}{x^2 + 4} \)?

**Tip:** Always check the relationship between the numerator and the derivative of the denominator when dealing with integrals of the form \( \frac{f(x)}{g(x)} \).

Learn how to find the integral of the function f(x)/F(x) using the substitution method. This detailed step-by-step solution covers integral calculus concepts such as antiderivatives and integration techniques. Suitable for advanced high school and college students.

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