If `int f(x) dx = f(x)` , then

Options

A. `f(x) = x`
B. `f(x)` = constant
C. `f(x) = 0`
D. `f(x) = e^x`

Answer

To solve the equation \( \int f(x) \, dx = f(x) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \int f(x) \, dx = f(x) \]
We can rearrange this to:
\[ \int f(x) \, dx - f(x) = 0 \]

### Step 2: Differentiate both sides
Next, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx} \left( \int f(x) \, dx \right) - \frac{d}{dx} (f(x)) = 0 \]
Using the Fundamental Theorem of Calculus, we know that:
\[ f(x) - f'(x) = 0 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ f'(x) = f(x) \]

### Step 4: Solve the differential equation
This is a standard first-order linear differential equation. The general solution is:
\[ f(x) = Ce^x \]
where \( C \) is a constant.

### Step 5: Conclusion
Thus, the function \( f(x) \) that satisfies the original equation is:
\[ f(x) = Ce^x \]