If `int(f(x))/(logsinx)dx=loglogsinx,` then f(x) is equal to

Options

A. sinx
B. cosx
C. logsinx
D. cotx

Answer

To solve the problem, we need to find the function \( f(x) \) given the equation:

\[ \int \frac{f(x)}{\log(\sin x)} \, dx = \log(\log(\sin x)) \]

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

We start by differentiating both sides of the equation to eliminate the integral. Using the Fundamental Theorem of Calculus, we have:

\[ \frac{d}{dx} \left( \int \frac{f(x)}{\log(\sin x)} \, dx \right) = \frac{f(x)}{\log(\sin x)} \]

On the right side, we differentiate \( \log(\log(\sin x)) \):

\[ \frac{d}{dx} \left( \log(\log(\sin x)) \right) = \frac{1}{\log(\sin x)} \cdot \frac{d}{dx}(\log(\sin x)) \]

### Step 2: Differentiate \( \log(\sin x) \)

Now, we need to differentiate \( \log(\sin x) \):

\[ \frac{d}{dx}(\log(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \cot x \]

### Step 3: Substitute back into the equation

Substituting this back into our equation, we have:

\[ \frac{f(x)}{\log(\sin x)} = \frac{\cot x}{\log(\sin x)} \]

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

Now, we can solve for \( f(x) \) by multiplying both sides by \( \log(\sin x) \):

\[ f(x) = \cot x \]

### Final Result

Thus, the function \( f(x) \) is:

\[ f(x) = \cot x \]