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:

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

where \(c\) is the constant of integration, and we know that \(f(1) = \frac{1}{6}\). We need to find \(f(6\pi)\).

### Step 1: Differentiate both sides

We differentiate both sides of the equation 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 get:

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

### Step 2: Rearranging the equation

We can rearrange this equation:

\[ f(x) - 6[f(x)]f'(x) = 0 \]

This can be factored as:

\[ f(x)(1 - 6f'(x)) = 0 \]

### Step 3: Solve the factors

From the factored equation, we have two cases:

1. \(f(x) = 0\)
2. \(1 - 6f'(x) = 0\)

Since \(f(x) = 0\) does not satisfy \(f(1) = \frac{1}{6}\), we focus on the second case:

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

### Step 4: Integrate \(f'(x)\)

Now we integrate \(f'(x)\):

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

### Step 5: Use the initial condition

We use the initial condition \(f(1) = \frac{1}{6}\) to find \(C\):

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

This simplifies to:

\[ \frac{1}{6} + C = \frac{1}{6} \implies C = 0 \]

### Step 6: Write the final function

Thus, we have:

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

### Step 7: Calculate \(f(6\pi)\)

Now we can find \(f(6\pi)\):

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

### Final Answer

Therefore, the value of \(f(6\pi)\) is:

\[ \boxed{\pi} \]

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