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To evaluate the integral 

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

using the integration by parts method, I'll outline the solution in 8 steps as requested.

---

### Steps for Solution

1. **Identify \( u \) and \( dv \)**:
   - We choose \( u = x^2 \) and \( dv = e^{-x} \, dx \).
   - This choice is based on the LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) which helps us decide the order for \( u \) and \( dv \).

2. **Differentiate \( u \) and Integrate \( dv \)**:
   - Differentiate \( u \) to find \( du \):  
     \[
     u = x^2 \Rightarrow du = 2x \, dx
     \]
   - Integrate \( dv \) to find \( v \):  
     \[
     dv = e^{-x} \, dx \Rightarrow v = -e^{-x}
     \]

3. **Apply Integration by Parts Formula**:
   - The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \).
   - Substitute \( u \), \( v \), \( dv \), and \( du \):
     \[
     \int x^2 e^{-x} \, dx = -x^2 e^{-x} - \int (-e^{-x}) (2x) \, dx
     \]
   - Simplify the expression:
     \[
     = -x^2 e^{-x} + 2 \int x e^{-x} \, dx
     \]

4. **Set Up Second Integration by Parts**:
   - For \( \int x e^{-x} \, dx \), we apply integration by parts again.
   - Choose \( u = x \) and \( dv = e^{-x} \, dx \).

5. **Differentiate \( u = x \) and Integrate \( dv = e^{-x} \, dx \)**:
   - Differentiate \( u \):  
     \[
     u = x \Rightarrow du = dx
     \]
   - Integrate \( dv \):  
     \[
     dv = e^{-x} \, dx \Rightarrow v = -e^{-x}
     \]

6. **Apply Integration by Parts Again**:
   - Substitute into the integration by parts formula:
     \[
     \int x e^{-x} \, dx = -x e^{-x} - \int (-e^{-x}) \, dx
     \]
   - Simplify:
     \[
     = -x e^{-x} + \int e^{-x} \, dx
     \]

7. **Evaluate the Remaining Integral**:
   - Solve \( \int e^{-x} \, dx \):
     \[
     \int e^{-x} \, dx = -e^{-x}
     \]
   - Substitute back:
     \[
     \int x e^{-x} \, dx = -x e^{-x} - e^{-x} = -(x + 1)e^{-x}
     \]

8. **Combine All Parts**:
   - Substitute back into the original integral:
     \[
     \int x^2 e^{-x} \, dx = -x^2 e^{-x} + 2(-(x + 1)e^{-x})
     \]
   - Simplify:
     \[
     = -x^2 e^{-x} - 2(x + 1)e^{-x}
     \]
   - Factor out \( e^{-x} \):
     \[
     = -(x^2 + 2x + 2)e^{-x}
     \]

---

### Final Answer

\[
\int x^2 e^{-x} \, dx = -(x^2 + 2x + 2)e^{-x} + C
\]

Would you like more details on any step?

---

### Related Questions

1. How does the LIATE rule help in choosing \( u \) and \( dv \) in integration by parts?
2. What are some alternative methods to evaluate integrals involving polynomials and exponentials?
3. How would the process change if the integrand was \( x^3 e^{-x} \) instead?
4. Why is integration by parts repeated when the \( u \)-term is still a polynomial?
5. How can you check your answer after performing integration by parts?

### Tip
For integration by parts, choose \( u \) such that it becomes simpler when differentiated, allowing \( dv \) to be easily integrated.

Evaluate ∫(x^2 e^{-x}) dx using integration by parts method. Number and describe all 8-steps, as covered during lecture.

This detailed solution explains how to evaluate the integral ∫(x^2 e^{-x}) dx using the integration by parts method, following 8 structured steps. Ideal for college calculus students learning about integration techniques involving polynomials and exponential functions.

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