To solve the integral \( I = \int (x - 1)e^{-x} \, dx \), we can break it down into simpler parts. Here’s a step-by-step solution:
### Step 1: Split the Integral
We can express the integral as:
\[
I = \int (x - 1)e^{-x} \, dx = \int x e^{-x} \, dx - \int e^{-x} \, dx
\]
### Step 2: Solve \( \int x e^{-x} \, dx \) using Integration by Parts
For the integral \( \int x e^{-x} \, dx \), we will use integration by parts. Let:
- \( u = x \) (thus \( du = dx \))
- \( dv = e^{-x} dx \) (thus \( v = -e^{-x} \))
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[
\int x e^{-x} \, dx = -x e^{-x} - \int (-e^{-x}) \, dx
\]
This simplifies to:
\[
\int x e^{-x} \, dx = -x e^{-x} + \int e^{-x} \, dx
\]
### Step 3: Solve \( \int e^{-x} \, dx \)
The integral \( \int e^{-x} \, dx \) is:
\[
\int e^{-x} \, dx = -e^{-x}
\]
### Step 4: Substitute Back
Now substituting back into our equation from Step 2:
\[
\int x e^{-x} \, dx = -x e^{-x} - e^{-x}
\]
### Step 5: Combine the Results
Now we can combine the results:
\[
I = \int x e^{-x} \, dx - \int e^{-x} \, dx = (-x e^{-x} - e^{-x}) - (-e^{-x})
\]
The \( -e^{-x} \) terms cancel out:
\[
I = -x e^{-x}
\]
### Step 6: Add the Constant of Integration
Finally, we add the constant of integration \( C \):
\[
I = -x e^{-x} + C
\]
Thus, the value of the integral \( \int (x - 1)e^{-x} \, dx \) is:
\[
\boxed{-x e^{-x} + C}
\]
