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

Answer

To solve the integral \( I = \int x^2 e^x \, dx \), 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 1: Choose \( u \) and \( dv \)

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

### Step 2: Differentiate \( u \) and Integrate \( dv \)

Now we differentiate \( u \) and integrate \( dv \):
- \( du = 2x \, dx \)
- \( v = \int e^x \, dx = e^x \)

### Step 3: Apply Integration by Parts

Using the integration by parts formula, we have:

\[ I = uv - \int v \, du \]

Substituting the values we found:

\[ I = x^2 e^x - \int e^x (2x) \, dx \]

### Step 4: Simplify the Integral

Now we need to evaluate the integral \( \int 2x e^x \, dx \). We will apply integration by parts again.

Let:
- \( u = 2x \)
- \( dv = e^x \, dx \)

Then,
- \( du = 2 \, dx \)
- \( v = e^x \)

### Step 5: Apply Integration by Parts Again

Using the integration by parts formula again:

\[ \int 2x e^x \, dx = 2x e^x - \int e^x (2) \, dx \]

This simplifies to:

\[ \int 2x e^x \, dx = 2x e^x - 2 \int e^x \, dx \]

Now, we know that \( \int e^x \, dx = e^x \), so:

\[ \int 2x e^x \, dx = 2x e^x - 2e^x \]

### Step 6: Substitute Back into the Original Integral

Now we substitute back into our expression for \( I \):

\[ I = x^2 e^x - (2x e^x - 2e^x) \]

This simplifies to:

\[ I = x^2 e^x - 2x e^x + 2e^x \]

### Step 7: Factor the Expression

We can factor out \( e^x \):

\[ I = e^x (x^2 - 2x + 2) + C \]

where \( C \) is the constant of integration.

### Final Answer

Thus, the integral is:

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

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