Question

If \(\int \frac{1}{f(x)}dx=\log |f(x)|^{2}+c\), then \(f(x)=\ldots\)

• A. \(x+c\)
• B. \(\frac{x}{2}+c\)
• C. \(\frac{x^{2}}{2}+c\)
• D. \(x^{2}+c\)

Hint:

Differentiate both sides of the given relation to get a simple equation for the derivative of f.

Answer 1

The Correct Option is B

Step 1: Differentiate the given equation
$\frac{1}{f}=\frac{d}{dx}\left(2\log|f|\right)=\frac{2f'}{f}$.

Step 2: Cancel f
This gives $f'(x)=\frac{1}{2}$.

Step 3: Compare the options
Only $f(x)=\frac{x}{2}+c$ has derivative $\frac{1}{2}$, the second option.

Final Answer:
The function must be x/2 + c. \[ \boxed{\text{(B) }\dfrac{x}{2}+c} \]