`intxe^(x)dx=?`

Options

A. `e^(x)(1-x)+C`
B. `e^(x)(x+1)+C`
C. `e^(x)(x-1)+C`
D. none of these

Explanation

To solve the integral \( \int x e^x \, dx \), we will use the method of 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 means \( du = dx \))
- \( dv = e^x \, dx \) (which means \( v = e^x \))

### Step 2: Apply the integration by parts formula

Now, we can apply the integration by parts formula:

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

Substituting the values we have:

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

### Step 3: Solve the remaining integral

The integral \( \int e^x \, dx \) is straightforward:

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

### Step 4: Substitute back into the equation

Now substituting this back into our equation:

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

where \( C \) is the constant of integration.

### Final Answer

Thus, the final result is:

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