`int e^x {f(x)-f'(x)}dx= phi(x)`, then `int e^x f(x) dx` is

Options

A. `phi(x)=e^(x)f(x)`
B. `phi(x)-e^(x)f(x)`
C. `(1)/(2){phi(x)+e^(x)f(x)}`
D. `(1)/(2){phi(x)+e^(x)f'(x)}`

Explanation

To solve the integral \( \int e^x (f(x) - f'(x)) \, dx = \phi(x) \), we need to find the value of \( \int e^x f(x) \, dx \).

### Step-by-Step Solution

1. **Start with the given equation:**
\[ \int e^x (f(x) - f'(x)) \, dx = \phi(x) \]

2. **Distribute the integral:**
\[ \int e^x f(x) \, dx - \int e^x f'(x) \, dx = \phi(x) \]

3. **Use integration by parts on the second integral \( \int e^x f'(x) \, dx \):**
- Let \( u = f'(x) \) and \( dv = e^x dx \).
- Then \( du = f''(x) dx \) and \( v = e^x \).
- By integration by parts:
\[ \int e^x f'(x) \, dx = e^x f'(x) - \int e^x f''(x) \, dx \]

4. **Substituting back into the equation:**
\[ \int e^x f(x) \, dx - \left( e^x f'(x) - \int e^x f''(x) \, dx \right) = \phi(x) \]

5. **Rearranging the equation:**
\[ \int e^x f(x) \, dx + \int e^x f''(x) \, dx = \phi(x) + e^x f'(x) \]

6. **Isolate \( \int e^x f(x) \, dx \):**
\[ \int e^x f(x) \, dx = \phi(x) + e^x f'(x) - \int e^x f''(x) \, dx \]

7. **Now, we can express \( \int e^x f(x) \, dx \) in terms of \( \phi(x) \):**
\[ \int e^x f(x) \, dx = \frac{1}{2} \left( \phi(x) + e^x f'(x) \right) \]

### Final Result:
Thus, the value of \( \int e^x f(x) \, dx \) is:
\[ \int e^x f(x) \, dx = \frac{1}{2} \left( \phi(x) + e^x f'(x) \right) \]