https://file.aiquickdraw.com/mathbot/file/1721988266698mqqtxz3r.jpg

The expression in the image is \( 4e^{3x} \).

To integrate this function with respect to \( x \), we need to perform the following steps:

\[
\int 4e^{3x} \, dx
\]

We can use the fact that the integral of \( e^{ax} \) with respect to \( x \) is \( \frac{1}{a} e^{ax} \), where \( a \) is a constant. Here, \( a = 3 \).

\[
\int 4e^{3x} \, dx = 4 \int e^{3x} \, dx
\]

Using the integral formula:

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

Thus,

\[
4 \int e^{3x} \, dx = 4 \cdot \frac{1}{3} e^{3x} = \frac{4}{3} e^{3x}
\]

So, the integral is:

\[
\int 4e^{3x} \, dx = \frac{4}{3} e^{3x} + C
\]

where \( C \) is the constant of integration.

Would you like any 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?
2. What is the general formula for the integral of \( e^{ax} \)?
3. Can you explain the concept of the constant of integration?
4. How do you integrate polynomial functions?
5. What are the basic rules of integration?

**Tip:** Always remember to include the constant of integration \( C \) when performing indefinite integrals.

Learn how to integrate 4e^{3x} with a detailed step-by-step solution. This problem covers integration of exponential functions, including the integral formula for e^{ax}. Suitable for advanced high school students and anyone studying calculus.

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