What is `int_0^1 xe^x dx` equal to

Options

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

Answer

To solve 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 means \( du = dx \))
- \( dv = e^x \, dx \) (which means \( v = e^x \))

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

\[ \int_0^1 x e^x \, dx = \left[ x e^x \right]_0^1 - \int_0^1 e^x \, dx \]

### Step 3: Evaluate \( \left[ x e^x \right]_0^1 \)
Now we evaluate \( \left[ x e^x \right]_0^1 \):

\[ \left[ x e^x \right]_0^1 = (1 \cdot e^1) - (0 \cdot e^0) = e - 0 = e \]

### Step 4: Evaluate \( \int_0^1 e^x \, dx \)
Next, we need to evaluate \( \int_0^1 e^x \, dx \):

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

Thus,

\[ \int_0^1 e^x \, dx = \left[ e^x \right]_0^1 = e^1 - e^0 = e - 1 \]

### Step 5: Substitute back into the equation
Now we substitute back into our equation:

\[ \int_0^1 x e^x \, dx = e - (e - 1) = e - e + 1 = 1 \]

### Final Answer
Thus, we find that:

\[ \int_0^1 x e^x \, dx = 1 \]