The integral in the image is:

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

Solution:

To solve this integral, let's apply the basic integration rule for exponential functions:

$\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C$

Here, $k = 8$. So, we need to adjust for the coefficient of 8 inside the exponential:

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

Using the rule for exponential integrals:

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

Now, multiplying by 4:

$4 \cdot \frac{1}{8} e^{8x} = \frac{4}{8} e^{8x} = \frac{1}{2} e^{8x}$

Thus, the integral is:

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

Answer:

The correct answer is Option C: $\frac{1}{2} e^{8x} + C$.

Would you like further details on this explanation or have any questions? Here are some related questions to expand on this topic:

1. How do you integrate functions with different exponential bases?
2. What is the general rule for integrating exponentials with linear terms in the exponent?
3. How would you differentiate an integral result like $\frac{1}{2} e^{8x} + C$?
4. Can you apply substitution to solve this integral? How?
5. What happens if we have a product of an exponential and another function (like a polynomial)?

Tip: Always check if you need to divide by the coefficient of $x$ when integrating exponential functions.