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The given integral is:

\[
\int x^2 e^{\frac{-x}{2}} \, dx
\]

We can solve this using integration by parts, with the following steps:

1. **Choose the parts**:
   Let \( u = x^2 \) and \( dv = e^{\frac{-x}{2}} dx \).

   - Differentiate \( u \): 
   \[
   du = 2x \, dx
   \]
   
   - Integrate \( dv \):
   \[
   v = \int e^{\frac{-x}{2}} \, dx = -2 e^{\frac{-x}{2}}
   \]

2. **Apply the integration by parts formula**:
   \[
   \int u \, dv = uv - \int v \, du
   \]

   Substituting the values:
   \[
   \int x^2 e^{\frac{-x}{2}} \, dx = -2x^2 e^{\frac{-x}{2}} + \int 2x \cdot (-2) e^{\frac{-x}{2}} \, dx
   \]
   Simplifying the integral:
   \[
   = -2x^2 e^{\frac{-x}{2}} + 4 \int x e^{\frac{-x}{2}} \, dx
   \]

3. **Solve the remaining integral**: 
   Use integration by parts again for \( \int x e^{\frac{-x}{2}} \, dx \), setting \( u = x \) and \( dv = e^{\frac{-x}{2}} dx \):
   
   - Differentiate \( u \): 
   \[
   du = dx
   \]
   
   - Integrate \( dv \): 
   \[
   v = -2 e^{\frac{-x}{2}}
   \]
   
   Apply the formula again:
   \[
   \int x e^{\frac{-x}{2}} \, dx = -2x e^{\frac{-x}{2}} + 2 \int e^{\frac{-x}{2}} \, dx
   \]
   
   The remaining integral is:
   \[
   \int e^{\frac{-x}{2}} \, dx = -2 e^{\frac{-x}{2}}
   \]
   
   Therefore:
   \[
   \int x e^{\frac{-x}{2}} \, dx = -2x e^{\frac{-x}{2}} - 4 e^{\frac{-x}{2}}
   \]

4. **Substitute this back into the original equation**:
   \[
   \int x^2 e^{\frac{-x}{2}} \, dx = -2x^2 e^{\frac{-x}{2}} + 4(-2x e^{\frac{-x}{2}} - 4 e^{\frac{-x}{2}})
   \]

   Simplifying:
   \[
   = -2x^2 e^{\frac{-x}{2}} - 8x e^{\frac{-x}{2}} - 16 e^{\frac{-x}{2}} + C
   \]

Thus, the solution to the integral is:

\[
\boxed{-2x^2 e^{\frac{-x}{2}} - 8x e^{\frac{-x}{2}} - 16 e^{\frac{-x}{2}} + C}
\]

Would you like further clarification on any part of this solution?

---

Here are 5 questions related to this topic:
1. How is integration by parts applied to integrals involving products of polynomials and exponentials?
2. Can you describe how the choice of \( u \) and \( dv \) affects the integration process?
3. How would the result change if the exponent of \( e \) was different, for example \( e^{-\frac{x}{3}} \)?
4. What would be the steps for solving \( \int x^3 e^{\frac{-x}{2}} \, dx \)?
5. How do you check if the integration by parts formula has been applied correctly?

**Tip**: When solving integrals using integration by parts, try to reduce the complexity of the integrand at each step by choosing the parts carefully.

Learn how to solve the integral ∫ x^2 e^(x/2) dx using integration by parts. This problem involves key calculus concepts, including the application of the integration by parts formula, and is suitable for students in Grade 11-12 or college level.

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