To integrate the function $x e^{-\frac{x}{2}}$ from $0$ to $2$, we can use integration by parts.

Let: $I = \int_{0}^{2} x e^{-\frac{x}{2}} \, dx$

Step 1: Set Up for Integration by Parts

Choose:

• $u = x$, so $du = dx$
• $dv = e^{-\frac{x}{2}} \, dx$, so $v = -2 e^{-\frac{x}{2}}$

Using the integration by parts formula: $\int u \, dv = uv - \int v \, du$

Step 2: Apply Integration by Parts

$I = \left[ -2x e^{-\frac{x}{2}} \right]_{0}^{2} + \int_{0}^{2} 2 e^{-\frac{x}{2}} \, dx$

Evaluating the Boundary Term

For the boundary term: $-2x e^{-\frac{x}{2}} \bigg|_{0}^{2} = -2 \cdot 2 \cdot e^{-1} - (-2 \cdot 0 \cdot e^{0}) = -4 e^{-1} = -\frac{4}{e}$

Evaluating the Integral Term

Now, we integrate $2 e^{-\frac{x}{2}}$: $\int_{0}^{2} 2 e^{-\frac{x}{2}} \, dx = -4 e^{-\frac{x}{2}} \bigg|_{0}^{2} = -4 e^{-1} + 4$

Step 3: Combine Results

Now we combine everything: $I = -\frac{4}{e} + (-4 e^{-1} + 4) = -\frac{8}{e} + 4$

So, the result is: $\int_{0}^{2} x e^{-\frac{x}{2}} \, dx = 4 - \frac{8}{e}$

Would you like further details, or do you have any questions?

Related Questions

1. How do you apply integration by parts to other types of functions?
2. What is the significance of using boundaries in definite integrals?
3. How would this integration process change if $e^{-\frac{x}{2}}$ had a different coefficient?
4. What are some alternative techniques for evaluating integrals with exponential functions?
5. How do we interpret the result of this integral in terms of areas under curves?

Tip: When integrating functions with exponentials, consider substitution or integration by parts, as these methods often simplify the process.