Compute the integral `int_(0)^(1) xe^(x) dx`

Answer

To compute the integral \( \int_{0}^{1} x e^{x} \, dx \), we can 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 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:

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

Substituting our choices for \( u \) and \( dv \):

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

### Step 3: Compute the integral of \( e^{x} \)

The integral of \( e^{x} \) is:

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

### Step 4: Substitute back into the equation

Now substituting back, we get:

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

### Step 5: 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 6: Combine the results

Now, we combine the results from the upper and lower limits:

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

Thus, the value of the integral is:

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