If `int(1)/(f(x))dx=log[f(x)]^(2)+c`, then `f(x)=`

Options

A. `x+a`
B. `2x+a`
C. `(x)/(2)+a`
D. `x^(2)+a`

Answer

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

\[ \int \frac{1}{f(x)} \, dx = \log[f(x)]^2 + c \]

We need to find the function \( f(x) \).

### Step 1: Differentiate both sides

To find \( f(x) \), we will differentiate both sides with respect to \( x \).

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

\[ \frac{1}{f(x)} = \frac{d}{dx} \left( \log[f(x)]^2 + c \right) \]

### Step 2: Apply the chain rule on the right side

Using the chain rule, the derivative of \( \log[f(x)]^2 \) is:

\[ \frac{d}{dx} \left( \log[f(x)]^2 \right) = \frac{2 \cdot f'(x)}{f(x)} \]

Thus, we can rewrite the equation as:

\[ \frac{1}{f(x)} = \frac{2 f'(x)}{f(x)} \]

### Step 3: Simplify the equation

Since \( f(x) \) is present in both sides, we can multiply both sides by \( f(x) \):

\[ 1 = 2 f'(x) \]

### Step 4: Solve for \( f'(x) \)

From the equation above, we can isolate \( f'(x) \):

\[ f'(x) = \frac{1}{2} \]

### Step 5: Integrate to find \( f(x) \)

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

\[ f(x) = \int \frac{1}{2} \, dx = \frac{1}{2} x + C \]

### Step 6: Write the final form of \( f(x) \)

Thus, we can express \( f(x) \) as:

\[ f(x) = \frac{1}{2} x + C \]

Where \( C \) is a constant of integration.

### Conclusion

The function \( f(x) \) that satisfies the original equation is:

\[ f(x) = \frac{1}{2} x + C \]