`int x^(2)e^(x)dx`

Answer

To solve the integral \( \int x^2 e^x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step-by-Step Solution:

1. **Choose \( u \) and \( dv \)**:
- Let \( u = x^2 \) (which we will differentiate)
- Let \( dv = e^x \, dx \) (which we will integrate)

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

3. **Apply the Integration by Parts Formula**:
\[ \int x^2 e^x \, dx = uv - \int v \, du \]
Substituting the values we found:
\[ = x^2 e^x - \int e^x (2x) \, dx \]
This simplifies to:
\[ = x^2 e^x - 2 \int x e^x \, dx \]

4. **Now, we need to solve \( \int x e^x \, dx \)** using integration by parts again:
- Let \( u = x \) and \( dv = e^x \, dx \)
- Then, \( du = dx \) and \( v = e^x \)

5. **Apply Integration by Parts Again**:
\[ \int x e^x \, dx = uv - \int v \, du \]
Substituting the values:
\[ = x e^x - \int e^x \, dx \]
The integral \( \int e^x \, dx = e^x \), so:
\[ = x e^x - e^x \]

6. **Substitute Back**:
Now we substitute back into our previous equation:
\[ \int x^2 e^x \, dx = x^2 e^x - 2(x e^x - e^x) \]
Simplifying this:
\[ = x^2 e^x - 2x e^x + 2e^x \]

7. **Final Result**:
\[ = (x^2 - 2x + 2)e^x + C \]
where \( C \) is the constant of integration.

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