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el valor exacto de la region sombreada es: ?

The problem involves calculating the exact value of the shaded area under the function \( f(x) = x \cdot e^{-2x} + 1 \) between the vertical lines \( x = 0 \) and \( x = 2 \). To find the area, we need to evaluate the definite integral of the function over this interval:

\[
\text{Area} = \int_{0}^{2} \left( x \cdot e^{-2x} + 1 \right) \, dx.
\]

Let's compute this integral step-by-step.The exact value of the shaded area is:

\[
\text{Area} = \frac{9}{4} - \frac{5 e^{-4}}{4}.
\]

Would you like a detailed step-by-step explanation of how this integral was calculated, or do you have any other questions?

Here are some related questions that might interest you:

1. How do you compute the integral of a function involving an exponential term?
2. What are some techniques for simplifying expressions involving exponential functions?
3. How can you approximate the numerical value of the shaded area?
4. What is the behavior of the function \( f(x) = x \cdot e^{-2x} + 1 \) as \( x \to \infty \)?
5. How does the area change if the limits of integration are altered?

**Tip:** When working with integrals involving exponential functions, substitution methods can often simplify the integration process.

El valor exacto de la región sombreada es: ?

This problem involves finding the exact value of the shaded region under the curve f(x) = x * e^{-2x} + 1, using definite integrals. The solution covers integral calculation and exponential function properties for limits between x = 0 and x = 2. Suitable for students in advanced high school or university-level calculus.

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