If `intf(x)dx=3[f(x)]^(2)+c` (where c is the constant of integration) and `f(1)=(1)/(6)`, then `f(6pi)` is equal to

Options

A. `(pi)/(2)`
B. `pi`
C. `(pi)/(3)`
D. `(pi)/(6)`

Answer

To solve the problem step by step, we start with the given equation:

\[ \int f(x) \, dx = 3[f(x)]^2 + c \]

### Step 1: Differentiate both sides
We differentiate both sides with respect to \(x\):

\[ \frac{d}{dx}\left(\int f(x) \, dx\right) = \frac{d}{dx}\left(3[f(x)]^2 + c\right) \]

Using the Fundamental Theorem of Calculus on the left side, we have:

\[ f(x) = 6f(x)f'(x) \]

### Step 2: Rearranging the equation
From the differentiation, we can rearrange the equation:

\[ f(x) = 6f(x)f'(x) \]

Dividing both sides by \(f(x)\) (assuming \(f(x) \neq 0\)) gives:

\[ 1 = 6f'(x) \]

### Step 3: Solve for \(f'(x)\)
Now, we can solve for \(f'(x)\):

\[ f'(x) = \frac{1}{6} \]

### Step 4: Integrate to find \(f(x)\)
Next, we integrate \(f'(x)\):

\[ f(x) = \int \frac{1}{6} \, dx = \frac{x}{6} + C \]

### Step 5: Use the initial condition
We know from the problem that \(f(1) = \frac{1}{6}\). We can use this to find the constant \(C\):

\[ f(1) = \frac{1}{6} + C = \frac{1}{6} \]

This implies:

\[ C = 0 \]

### Step 6: Final form of \(f(x)\)
Thus, the function simplifies to:

\[ f(x) = \frac{x}{6} \]

### Step 7: Find \(f(6\pi)\)
Now we need to find \(f(6\pi)\):

\[ f(6\pi) = \frac{6\pi}{6} = \pi \]

### Conclusion
The value of \(f(6\pi)\) is:

\[ \boxed{\pi} \]