`int_(0)^(1)x e^(x)dx=`

Options

A. 1
B. `e^(2)`
C. `1+e^(2)`
D. 2

Answer

To solve the integral \( \int_{0}^{1} x e^{x} \, dx \), we will use 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:
- \( u = x \) (which implies \( du = dx \))
- \( dv = e^{x} \, dx \) (which implies \( v = e^{x} \))

### Step 2: Apply the integration by parts formula
Using the integration by parts formula, we have:

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

### Step 3: Integrate \( \int e^{x} \, dx \)
The integral of \( e^{x} \) is simply \( e^{x} \):

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

### Step 4: Substitute back into the equation
Now substituting back into our equation:

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

### Step 5: Evaluate from 0 to 1
Now we need to evaluate this expression from 0 to 1:

\[ \left[ x e^{x} - e^{x} \right]_{0}^{1} \]

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

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

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

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

### Step 6: Combine the results
Now, we combine the results from the upper and lower limits:

\[ \left( 0 \right) - \left( -1 \right) = 0 + 1 = 1 \]

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

\[ \boxed{1} \]