`int sqrt(1+sinx )(cos ""(x)/(2)-sin ""(x)/(2))` dx is:

Options

A. A) `(1+ sinx)/(2) +C`
B. B) `(1+ sin x)^(2)`
C. C) `(1)/(sqrt(1+sin x)) +C`
D. D) `sin x+C`

Answer

To solve the integral \( I = \int \sqrt{1 + \sin x} \left( \frac{\cos x}{2} - \frac{\sin x}{2} \right) dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \sqrt{1 + \sin x} \left( \frac{\cos x}{2} - \frac{\sin x}{2} \right) dx \]

### Step 2: Simplify \( \sqrt{1 + \sin x} \)
Using the identity \( 1 + \sin x = \cos^2\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) + 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) \), we can express:
\[ 1 + \sin x = \left( \cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) \right)^2 \]
Thus,
\[ \sqrt{1 + \sin x} = \cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) \]

### Step 3: Substitute Back into the Integral
Substituting this back into the integral, we have:
\[ I = \int \left( \cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) \right) \left( \frac{\cos x}{2} - \frac{\sin x}{2} \right) dx \]

### Step 4: Expand the Integral
Now we expand the expression:
\[ I = \int \left( \frac{\cos\left(\frac{x}{2}\right) \cos x}{2} - \frac{\cos\left(\frac{x}{2}\right) \sin x}{2} + \frac{\sin\left(\frac{x}{2}\right) \cos x}{2} - \frac{\sin\left(\frac{x}{2}\right) \sin x}{2} \right) dx \]

### Step 5: Use Trigonometric Identities
Using the identity \( \cos^2 a - \sin^2 a = \cos 2a \), we can combine terms:
\[ I = \int \left( \frac{1}{2} \left( \cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right) \right) \right) dx \]
This simplifies to:
\[ I = \frac{1}{2} \int \cos x \, dx \]

### Step 6: Integrate
Integrating \( \cos x \):
\[ I = \frac{1}{2} \sin x + C \]

### Step 7: Final Result
Thus, the final result of the integral is:
\[ I = \sin x + C \]

### Conclusion
The correct answer is \( \sin x + C \), which corresponds to option D.