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.