The value of `int_(0)^(1)x^(2)e^(x)dx` is equal to

Options

A. `e-2`
B. `e+2`
C. `e^(2)-2`
D. `e^(2)`

Answer

To solve the integral \( I = \int_{0}^{1} x^2 e^x \, dx \), 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 and Integrate

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

### Step 3: Apply Integration by Parts

Using the integration by parts formula:

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

Substituting the values we have:

\[ I = x^2 e^x \bigg|_{0}^{1} - \int_{0}^{1} e^x (2x) \, dx \]

### Step 4: Evaluate the Boundary Terms

Now, we evaluate \( x^2 e^x \) at the limits 0 and 1:

\[ x^2 e^x \bigg|_{0}^{1} = (1^2 e^1) - (0^2 e^0) = e - 0 = e \]

### Step 5: Simplify the Integral

Now we need to compute the integral \( \int_{0}^{1} 2x e^x \, dx \):

\[ I = e - 2 \int_{0}^{1} x e^x \, dx \]

### Step 6: Apply Integration by Parts Again

For the integral \( \int x e^x \, dx \), we apply integration by parts again:
- Let \( u = x \) and \( dv = e^x \, dx \)
- Then \( du = dx \) and \( v = e^x \)

Using integration by parts:

\[ \int x e^x \, dx = x e^x \bigg|_{0}^{1} - \int e^x \, dx \]

Evaluating the boundary terms:

\[ x e^x \bigg|_{0}^{1} = (1 e^1) - (0 e^0) = e - 0 = e \]

Now, compute the remaining integral:

\[ \int e^x \, dx = e^x \bigg|_{0}^{1} = e - 1 \]

So we have:

\[ \int x e^x \, dx = e - (e - 1) = 1 \]

### Step 7: Substitute Back

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

\[ I = e - 2 \cdot 1 = e - 2 \]

### Final Answer

Thus, the value of the integral \( \int_{0}^{1} x^2 e^x \, dx \) is:

\[ \boxed{e - 2} \]