The value of `int(x-1)e^(-x)` dx is equal to

Options

A. `-xe^(x)+C`
B. `xe^(x)+C`
C. `-xe^(-x)+C`
D. `xe^(-x)+C`

Answer

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:
\[ \boxed{-x e^{-x} + C} \]