To integrate the function $2e^x \, dx$:

Steps:

1. Recognize that $e^x$ is its own derivative and integral.
2. Use the constant multiple rule: $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.

Solution:

$\int 2e^x \, dx = 2 \int e^x \, dx$ $\int e^x \, dx = e^x + C$ $\therefore \int 2e^x \, dx = 2e^x + C$

Final Answer:

$2e^x + C$

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Would you like a detailed explanation of any step or have any questions? Here are some follow-up questions to expand your understanding:

1. What is the constant of integration, and why is it necessary?
2. How is $e^x$ unique compared to other functions in terms of differentiation and integration?
3. How does the constant multiple rule apply in integral calculus?
4. Can you provide another example of integrating exponential functions?
5. What is the derivative of $2e^x + C$ to verify the solution?

Tip: Always double-check your results by differentiating your integral to ensure you get back the original function.