Question

If \(y = \sqrt{cosx^2+\sqrt{cosx^2+\sqrt{cosx^2+\ldots \infty }}}\) and \(\frac{dy}{dx} = \frac{f(x)}{2y-1}\) then, \(\int f(x)\,dx = \ldots\)

• A. \(sinx^2+c\)
• B. \(-sinx^2+c\)
• C. \(cosx^2+c\)
• D. \(-cosx^2+c\)

Hint:

Use the chain rule and the derivative of log((x+3)/x).

Answer 1

The Correct Option is C

Step 1: Quotient rule on the inner function:
$g = \frac{x + 3}{x} = 1 + \frac3x$, so $g' = -\frac{3}{x^2}$.

Step 2: Combine:
$\frac{d}{dx}\log g = \frac{g'}{g} = -\frac{3}{x^2}\cdot\frac{x}{x + 3} = \frac{-3}{x(x + 3)}$. Multiplying by $\cos(\log g)$ gives option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C)}} \]