Evaluate the following :
`int_(0)^(1) xe^(x) dx`

Answer

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

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

### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = x \) (which is an algebraic function)
- \( dv = e^{x} \, dx \) (which is an exponential function)

### Step 2: Differentiate \( u \) and integrate \( dv \)
Now, we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = dx \]
- Integrate \( dv \):
\[ v = \int e^{x} \, dx = e^{x} \]

### Step 3: Apply the integration by parts formula
Now, substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:
\[ \int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx \]

### Step 4: Evaluate the integral
Now we need to evaluate \( \int e^{x} \, dx \):
\[ \int e^{x} \, dx = e^{x} \]
So, substituting back, we have:
\[ \int x e^{x} \, dx = x e^{x} - e^{x} + C \]

### Step 5: Evaluate the definite integral from 0 to 1
Now we evaluate the definite integral from 0 to 1:
\[ \int_{0}^{1} x e^{x} \, dx = \left[ x e^{x} - e^{x} \right]_{0}^{1} \]

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, substituting the limits:
\[ \int_{0}^{1} x e^{x} \, dx = [0 - (-1)] = 0 + 1 = 1 \]

Thus, the final result is:
\[ \int_{0}^{1} x e^{x} \, dx = 1 \]