What is `int_(0)^(1) xe^(x) dx` equal to

Options

A. 1
B. -1
C. 0
D. e

Explanation

To solve the integral \( \int_{0}^{1} x e^{x} \, dx \), we will use integration by parts. The integration by parts formula is given by:

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

### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = x \) (which means \( du = dx \))
- \( dv = e^{x} \, dx \) (which means \( v = e^{x} \))

### Step 2: Apply the integration by parts formula
Now, applying the integration by parts formula:

\[ \int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx \]

### Step 3: Compute the integral \( \int e^{x} \, dx \)
The integral of \( e^{x} \) is simply \( e^{x} \). Therefore, we have:

\[ \int x e^{x} \, dx = x e^{x} - e^{x} + C \]

### Step 4: Evaluate the definite integral from 0 to 1
Now we need to evaluate this from 0 to 1:

\[ \int_{0}^{1} x e^{x} \, dx = \left[ x e^{x} - e^{x} \right]_{0}^{1} \]

Calculating the upper limit (when \( x = 1 \)):

\[ 1 \cdot e^{1} - e^{1} = e - e = 0 \]

Calculating the lower limit (when \( x = 0 \)):

\[ 0 \cdot e^{0} - e^{0} = 0 - 1 = -1 \]

### Step 5: Combine the results
Now, substituting back into the definite integral:

\[ \int_{0}^{1} x e^{x} \, dx = 0 - (-1) = 1 \]

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

\[ \boxed{1} \]