Read the following text and answer the followig questions on the basis of the same :
`inte^(x)[f(x) + f'(x)]dx = int e^(x)f(x)dx + int e^(x) f'(x)dx`
`= f(x)e^(x) - int f'(x)e^(x)dx + int f'(x)e^(x)dx`
`= e^(x)f(x) + c`
`int e^(x)(x+1)dx` = __________.

Options

A. `x e^(x) + c`
B. `e^(x) + c`
C. `e^(-x) + c`
D. None of these

Answer

To solve the integral \(\int e^{x}(x+1)dx\), we can use the formula derived from the integration of \(e^{x}[f(x) + f'(x)]dx\).

### Step-by-Step Solution:

1. **Identify \(f(x)\)**:
We can let \(f(x) = x\). Therefore, we need to find \(f'(x)\):
\[ f'(x) = \frac{d}{dx}(x) = 1 \]

2. **Apply the formula**:
According to the formula given, we have:
\[ \int e^{x}[f(x) + f'(x)]dx = e^{x}f(x) + C \]
Substituting \(f(x) = x\) and \(f'(x) = 1\):
\[ \int e^{x}(x + 1)dx = e^{x}x + C \]

3. **Final result**:
Thus, we can write the final result as:
\[ \int e^{x}(x + 1)dx = e^{x}x + C \]