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 \( \int (x - 1)e^{-x} \, dx \), we can use the method of integration by parts. Let's break it down step by step.

### Step 1: Identify \( u \) and \( dv \)
We will choose:
- \( u = x - 1 \) (which we will differentiate)
- \( dv = e^{-x} \, dx \) (which we will integrate)

### Step 2: Differentiate \( u \) and Integrate \( dv \)
Now we find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = dx \]
- Integrate \( dv \):
\[ v = \int e^{-x} \, dx = -e^{-x} \]

### Step 3: Apply the Integration by Parts Formula
The integration by parts formula is:
\[ \int u \, dv = uv - \int v \, du \]
Substituting our values:
\[ \int (x - 1)e^{-x} \, dx = (x - 1)(-e^{-x}) - \int (-e^{-x}) \, dx \]
This simplifies to:
\[ = -(x - 1)e^{-x} + \int e^{-x} \, dx \]

### Step 4: Integrate \( e^{-x} \)
We already know:
\[ \int e^{-x} \, dx = -e^{-x} \]
So we substitute this back into our equation:
\[ = -(x - 1)e^{-x} - e^{-x} \]

### Step 5: Combine Like Terms
Now we can simplify:
\[ = -(x - 1)e^{-x} - e^{-x} = -xe^{-x} + e^{-x} - e^{-x} = -xe^{-x} \]

### Step 6: Add the Constant of Integration
Finally, we include the constant of integration \( C \):
\[ \int (x - 1)e^{-x} \, dx = -xe^{-x} + C \]

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