This image contains a question about finding the indefinite integral of the function $e^{3x}$. Let’s solve this step by step.

We are tasked with finding the indefinite integral:

$\int e^{3x} \, dx$

To solve this, we use the substitution method. Let:

$u = 3x$

Thus, $du = 3dx$, or equivalently, $dx = \frac{du}{3}$.

Now, substituting into the integral:

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

The integral of $e^u$ is simply $e^u$, so we have:

$\frac{1}{3} e^u + C = \frac{1}{3} e^{3x} + C$

Thus, the correct answer is:

$\boxed{\frac{1}{3} e^{3x} + C}$

From the provided options, this corresponds to option 3:

$\frac{1}{3} e^{3x} + C$

Let me know if you'd like further details or if you have more questions!

Here are some related questions for deeper exploration:

1. What is the general formula for integrating exponential functions like $e^{ax}$?
2. How does substitution work in integrals? Could you give another example?
3. How do you handle indefinite integrals with constants, like the $+ C$ term?
4. What are some common mistakes students make when solving integrals involving exponential functions?
5. How would you solve the integral $\int e^{2x} \, dx$ using a similar method?

Tip: Always check the substitution carefully; getting the differential term right (like $dx$) is key to a correct solution!