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The problem on the board asks to evaluate the integral \( \int x^2 e^x \, dx \) using the method of integration by parts.

### Step-by-Step Solution:

1. **Identify Parts for Integration by Parts Formula:**
   The integration by parts formula is:
   \[
   \int u \, dv = uv - \int v \, du
   \]
   We need to choose \( u \) and \( dv \) wisely:
   - Let \( u = x^2 \) (since the polynomial will reduce when we differentiate it).
   - Let \( dv = e^x \, dx \) (since integrating \( e^x \) is simple).

2. **Differentiate \( u \) and Integrate \( dv \):**
   - \( u = x^2 \), so \( du = 2x \, dx \).
   - \( dv = e^x \, dx \), so \( v = e^x \).

3. **Apply the Integration by Parts Formula:**
   Using the formula \( \int u \, dv = uv - \int v \, du \):
   \[
   \int x^2 e^x \, dx = x^2 e^x - \int 2x e^x \, dx
   \]
   Now, we need to evaluate the remaining integral \( \int 2x e^x \, dx \).

4. **Second Integration by Parts:**
   To solve \( \int 2x e^x \, dx \), apply integration by parts again:
   - Let \( u = 2x \) and \( dv = e^x \, dx \).
   - \( du = 2 \, dx \) and \( v = e^x \).
   
   Applying the formula again:
   \[
   \int 2x e^x \, dx = 2x e^x - \int 2 e^x \, dx
   \]
   The integral \( \int 2 e^x \, dx \) is easy to compute:
   \[
   \int 2 e^x \, dx = 2 e^x
   \]
   So,
   \[
   \int 2x e^x \, dx = 2x e^x - 2 e^x
   \]

5. **Substitute Back:**
   Now substitute back into the expression:
   \[
   \int x^2 e^x \, dx = x^2 e^x - (2x e^x - 2 e^x)
   \]
   Simplifying:
   \[
   \int x^2 e^x \, dx = x^2 e^x - 2x e^x + 2 e^x + C
   \]
   where \( C \) is the constant of integration.

### Final Answer:
\[
\int x^2 e^x \, dx = (x^2 - 2x + 2) e^x + C
\]

Would you like more details or clarifications on any step?

### Questions for Further Practice:
1. How does the integration by parts method handle products of different types of functions, such as polynomials and exponentials?
2. How would you evaluate \( \int x^3 e^x \, dx \)?
3. Can integration by parts be used to solve trigonometric integrals like \( \int x \sin(x) \, dx \)?
4. What is the result of differentiating the final answer to check the solution?
5. How would you modify the approach for an integral like \( \int x^n e^x \, dx \) for any \( n \)?

**Tip:** When choosing \( u \) in integration by parts, it's generally helpful to select a function that simplifies upon differentiation, such as a polynomial, to make the remaining integral easier to handle.

Use integration by parts to evaluate ∫ x^2 e^x dx. Number & describe each step, including the checking answer step.

Learn how to evaluate the integral ∫ x^2 e^x dx using the method of integration by parts. This problem involves key calculus concepts such as polynomial and exponential functions, and follows a step-by-step application of the integration by parts formula.

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