The problem asks to evaluate the integral:

$\int 4e^{8x} \, dx$

To solve this, we can follow these steps:

1. Recognize the integral form: The integral of an exponential function, $\int e^{ax} \, dx$, is $\frac{1}{a}e^{ax} + C$.

2. Apply substitution: For $4e^{8x}$, we identify $a = 8$.

3. Integrate:

   $\int 4e^{8x} \, dx = 4 \cdot \frac{1}{8} e^{8x} + C = \frac{1}{2} e^{8x} + C$

Thus, the correct answer is:

$\frac{1}{2} e^{8x} + C$

This matches option (C).

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:

1. How would you approach integrating a function of the form $e^{ax + b}$?
2. How does the constant factor $4$ influence the integration process?
3. What would change if the exponent were a function of $x$, such as $e^{x^2}$?
4. How is integration of exponential functions used in real-world applications?
5. What if the integral had limits, say $\int_0^1 4e^{8x} \, dx$?

Tip: When integrating exponential functions, identifying and factoring out constants helps simplify the process.