It appears you want to integrate the given function $4e^{3x}$. Here is the integration step-by-step:

1. Given Integral: $\int 4e^{3x} \, dx$

2. Extract the constant outside the integral: $4 \int e^{3x} \, dx$

3. Integral of $e^{3x}$: Recall that $\int e^{ax} \, dx = \frac{1}{a} e^{ax}$ where $a$ is a constant. In this case, $a = 3$.

   $\int e^{3x} \, dx = \frac{1}{3} e^{3x}$

4. Multiply by the constant extracted earlier: $4 \cdot \frac{1}{3} e^{3x} = \frac{4}{3} e^{3x}$

5. Add the constant of integration $C$: $\int 4e^{3x} \, dx = \frac{4}{3} e^{3x} + C$

Thus, the integral of $4e^{3x}$ with respect to $x$ is:

$\boxed{\frac{4}{3} e^{3x} + C}$

Would you like more details or have any questions about this integration?

Here are some related questions you might want to ask:

1. How do you integrate functions with exponential terms involving different bases?
2. Can you explain the process of integrating trigonometric functions?
3. How do you handle integrals involving polynomial and exponential terms?
4. What are the methods for solving definite integrals?
5. Can you explain integration by parts?

Tip: When integrating functions, always check if any constants can be factored out to simplify the integral before proceeding with the integration.