If `intf(x)dx=f(x)+c," then " f(x)=`

Options

A. `x^( e)`
B. `e^( x)`
C. `log|x|`
D. `(1)/(x)`

Answer

To solve the problem, we need to find the function \( f(x) \) given that:

\[ \int f(x) \, dx = f(x) + c \]

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \int f(x) \, dx = f(x) + c \]

2. **Differentiate both sides with respect to \( x \):**
\[ \frac{d}{dx} \left( \int f(x) \, dx \right) = \frac{d}{dx} (f(x) + c) \]

3. **Apply the Fundamental Theorem of Calculus on the left side:**
\[ f(x) = f'(x) + 0 \]
(since the derivative of a constant \( c \) is 0)

4. **Rearranging the equation gives:**
\[ f'(x) = f(x) \]

5. **This is a first-order linear differential equation. The general solution to this equation is:**
\[ f(x) = Ce^x \]
where \( C \) is a constant.

6. **To find a specific solution, we can assume \( C = 1 \) (as the problem does not specify any particular initial condition):**
\[ f(x) = e^x \]

### Conclusion:
Thus, the function \( f(x) \) is:

\[ f(x) = e^x \]