Compute the integrals :
`int (x - 1)e^(-x) dx`

Answer

To compute the integral \(\int (x - 1)e^{-x} \, dx\), 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:
- \(u = x - 1\) (which we will differentiate)
- \(dv = e^{-x} \, dx\) (which we will integrate)

### Step 2: Differentiate \(u\) and integrate \(dv\)

Now, we differentiate \(u\) and integrate \(dv\):
- \(du = dx\)
- To find \(v\), we integrate \(dv\):
\[ v = \int e^{-x} \, dx = -e^{-x} \]

### Step 3: Apply the integration by parts formula

Substituting \(u\), \(du\), \(v\), and \(dv\) into the integration by parts formula:
\[ \int (x - 1)e^{-x} \, dx = (x - 1)(-e^{-x}) - \int (-e^{-x}) \, dx \]

This simplifies to:
\[ = -(x - 1)e^{-x} + \int e^{-x} \, dx \]

### Step 4: Integrate \(e^{-x}\)

Now we compute the remaining integral:
\[ \int e^{-x} \, dx = -e^{-x} \]

### Step 5: Substitute back into the equation

Substituting this back into our equation gives:
\[ \int (x - 1)e^{-x} \, dx = -(x - 1)e^{-x} - e^{-x} + C \]

### Step 6: Simplify the expression

Now, we can simplify the expression:
\[ = -xe^{-x} + e^{-x} - e^{-x} + C \]
\[ = -xe^{-x} + C \]

### Final Answer

Thus, the final result of the integral is:
\[ \int (x - 1)e^{-x} \, dx = -xe^{-x} + C \]