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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?

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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.

Evaluate the integral \( \int 4e^{8x} \, dx \).

Learn how to evaluate the integral of the exponential function \( \int 4e^{8x} \, dx \). This solution covers key calculus concepts in integration, specifically applying the formula for exponential functions, and is suitable for advanced high school or introductory college calculus students.

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