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.