`"If " int sqrt(1+sinx) f(x)dx=(2)/(3)(1+sinx)^(3//2)+c, " then " f(x) " equals "`

Options

A. `cos x`
B. `sin x`
C. `tan x`
D. `1`

Explanation

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

\[ \int \sqrt{1 + \sin x} \, f(x) \, dx = \frac{2}{3} (1 + \sin x)^{\frac{3}{2}} + c \]

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

We start by differentiating both sides of the equation. Using the Fundamental Theorem of Calculus, we differentiate the left-hand side:

\[ \frac{d}{dx} \left( \int \sqrt{1 + \sin x} \, f(x) \, dx \right) = \sqrt{1 + \sin x} \, f(x) \]

For the right-hand side, we apply the chain rule:

\[ \frac{d}{dx} \left( \frac{2}{3} (1 + \sin x)^{\frac{3}{2}} \right) = \frac{2}{3} \cdot \frac{3}{2} (1 + \sin x)^{\frac{1}{2}} \cdot \cos x \]

This simplifies to:

\[ (1 + \sin x)^{\frac{1}{2}} \cdot \cos x \]

### Step 2: Set the derivatives equal to each other

Now we equate the derivatives obtained from both sides:

\[ \sqrt{1 + \sin x} \, f(x) = (1 + \sin x)^{\frac{1}{2}} \cdot \cos x \]

### Step 3: Solve for \( f(x) \)

To isolate \( f(x) \), we divide both sides by \( \sqrt{1 + \sin x} \):

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

### Final Answer

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

\[ \boxed{\cos x} \]