The integral you're evaluating is:

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

To explain why the answer is $3e^{x/3} + C$, let's break it down step by step.

Step 1: Simplify the integral

The given integral can be written as:

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

Since $\frac{1}{3}$ is a constant, you can factor it out of the integral:

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

Step 2: Integrate $e^x$

The integral of $e^x$ with respect to $x$ is simply $e^x$, because the exponential function $e^x$ is its own derivative:

$\int e^x \, dx = e^x + C_1$

where $C_1$ is the constant of integration.

Step 3: Multiply by the constant

Now, multiply the result of the integral by the constant $\frac{1}{3}$:

$\frac{1}{3} \cdot (e^x + C_1) = \frac{1}{3} e^x + \frac{1}{3} C_1$

Step 4: Simplify the constant

The constant term $\frac{1}{3} C_1$ can be simplified to just $C$, since any constant times a constant is still a constant. So, we can rewrite the answer as:

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

This is the result of the integral $\frac{e^x}{3} dx$.

However, it seems there may have been a misunderstanding in interpreting the integral you provided as leading to $3e^{x/3} + C$. If the original problem had been $\int e^{x/3} \, dx$, we would approach it differently, using substitution. Let me know if this might have been the case, and I can clarify further.

Do you have any questions or need further clarification on this explanation?

Here are some related questions that might interest you:

1. What is the integral of $e^{kx}$ where $k$ is a constant?
2. How does substitution work in integration, especially for functions like $e^{x/3}$?
3. Why do constants of integration combine into a single constant?
4. What are the general rules for integrating exponential functions?
5. How does integration by parts apply to more complex functions involving exponentials?

Tip: Always watch for constants in integrals, as they can factor in and simplify the problem early on!