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The problem asks to find the general antiderivative of the function:

\[
f(x) = 3e^{4x}
\]

### Solution:

To find the general antiderivative (indefinite integral) of the function \( f(x) = 3e^{4x} \), we perform the following steps:

1. **Identify the integral to solve:**
   \[
   \int 3e^{4x} \, dx
   \]

2. **Apply the constant multiple rule:**  
   The integral of a constant times a function is the constant times the integral of the function.
   \[
   3 \int e^{4x} \, dx
   \]

3. **Integrate the exponential function:**  
   The integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} \) where \( a \) is a constant.
   \[
   \int e^{4x} \, dx = \frac{1}{4} e^{4x}
   \]

4. **Multiply by the constant:**
   \[
   3 \cdot \frac{1}{4} e^{4x} = \frac{3}{4} e^{4x}
   \]

5. **Include the constant of integration \( C \):**
   \[
   \frac{3}{4} e^{4x} + C
   \]

### Final Answer:
The general antiderivative is:
\[
\frac{3}{4} e^{4x} + C
\]

Let me know if you want more details or if you have any questions!

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Here are some related questions you might find useful:

1. How do you find the antiderivative of a polynomial function?
2. What is the integral of \( e^{ax} \) in general terms?
3. How do you apply the constant multiple rule in integration?
4. What is the difference between indefinite and definite integrals?
5. How can you verify the antiderivative by differentiation?
6. What are the common antiderivatives of basic functions?
7. How do you solve integrals that involve substitution?
8. What are some practical applications of exponential integrals in real life?

**Tip:** Always remember to include the constant of integration \( C \) when finding indefinite integrals, as it represents the family of all possible antiderivatives.

Learn how to find the general antiderivative of the function f(x) = 3e^(4x) with detailed steps. This problem covers integration techniques, including the constant multiple rule and integrating exponential functions. Suitable for advanced high school students and beyond.

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