If `intf(x)dx=f(x),` then `int{f(x)}^2dx` is equal to

Options

A. `1/2 {f(x)}^(2)`
B. `{f(x)}^(3)`
C. `({f(x)}^(3))/(3)`
D. `{f(x)}^(2)`

Answer

To solve the problem, we start with the given information:

1. We know that \( \int f(x) \, dx = f(x) \).

This implies that differentiating \( f(x) \) with respect to \( x \) gives us \( f(x) \) itself:
\[ \frac{d}{dx} f(x) = f(x) \]

2. We need to find \( \int [f(x)]^2 \, dx \).

### Step-by-step Solution:

**Step 1: Use Integration by Parts**

We can use integration by parts, where we let:
- \( u = f(x) \) (first function)
- \( dv = f(x) \, dx \) (second function)

Then we differentiate and integrate:
- \( du = f'(x) \, dx \)
- \( v = f(x) \)

By the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we have:
\[ \int f(x) \cdot f(x) \, dx = f(x) \cdot f(x) - \int f(x) \cdot f'(x) \, dx \]

**Step 2: Substitute the Known Derivative**

From our earlier deduction, we know that \( f'(x) = f(x) \). Therefore, we can substitute this into our equation:
\[ \int f(x)^2 \, dx = f(x)^2 - \int f(x) \cdot f(x) \, dx \]
This simplifies to:
\[ \int f(x)^2 \, dx = f(x)^2 - \int f(x)^2 \, dx \]

**Step 3: Rearranging the Equation**

Now, we can rearrange the equation to isolate the integral:
\[ \int f(x)^2 \, dx + \int f(x)^2 \, dx = f(x)^2 \]
This gives:
\[ 2 \int f(x)^2 \, dx = f(x)^2 \]

**Step 4: Solve for the Integral**

Dividing both sides by 2, we find:
\[ \int f(x)^2 \, dx = \frac{1}{2} f(x)^2 \]

### Final Result

Thus, the final answer is:
\[ \int f(x)^2 \, dx = \frac{1}{2} f(x)^2 + C \]
where \( C \) is the constant of integration.