To solve the integral:

$\int 3e^{x+2} \, dx$

Step 1: Simplify the exponential expression.

Notice that the exponent $x + 2$ can be rewritten as $e^x \cdot e^2$. Since $e^2$ is just a constant, you can factor it out of the integral:

$\int 3e^{x+2} \, dx = 3e^2 \int e^x \, dx$

Step 2: Integrate $e^x$.

The integral of $e^x$ with respect to $x$ is simply $e^x$.

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

Step 3: Multiply by the constant $3e^2$.

Now, put everything back together:

$3e^2 \int e^x \, dx = 3e^2 e^x$

Final Answer:

$\int 3e^{x+2} \, dx = 3e^2 e^x + C$

where $C$ is the constant of integration.

Would you like further details on the process or any related questions?

Here are 5 follow-up questions:

1. How would the result change if the exponent was a linear function of $x$, like $2x + 5$?
2. How do you handle integrals involving constants multiplied with an exponential function?
3. Can you explain how to solve integrals with powers of $e$, such as $e^{2x}$?
4. What are the rules for integrating exponential functions with bases other than $e$?
5. How do you solve definite integrals involving exponential functions?

Tip: Always remember to factor out constants when dealing with exponentials—this makes your work simpler and more manageable.