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.