`int(sin x)/( sqrt(1+cos x))dx=?`

Options

A. `sqrt(1+cos x) +c`
B. `-2sqrt(1+ cos x)+c`
C. `2sqrt(1+ cos x) +c`
D. `2(1+ cos x) +c`

Answer

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

### Step 1: Simplify the Integral
We start with the integral:
\[ I = \int \frac{\sin x}{\sqrt{1 + \cos x}} \, dx \]

### Step 2: Use a Trigonometric Identity
Recall the identity for \( \sin x \):
\[ \sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} \]
Also, we know that:
\[ 1 + \cos x = 2 \cos^2 \frac{x}{2} \]
Thus, we can rewrite the integral:
\[ I = \int \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{\sqrt{2 \cos^2 \frac{x}{2}}} \, dx \]
This simplifies to:
\[ I = \int \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{\sqrt{2} \cos \frac{x}{2}} \, dx \]
Cancelling \( \cos \frac{x}{2} \) gives:
\[ I = \frac{2}{\sqrt{2}} \int \sin \frac{x}{2} \, dx \]

### Step 3: Factor Out Constants
The constant \( \frac{2}{\sqrt{2}} \) can be simplified to \( \sqrt{2} \):
\[ I = \sqrt{2} \int \sin \frac{x}{2} \, dx \]

### Step 4: Perform the Integral
Now we integrate \( \sin \frac{x}{2} \):
\[ \int \sin \frac{x}{2} \, dx = -2 \cos \frac{x}{2} + C \]
Thus, substituting back, we have:
\[ I = \sqrt{2} \left( -2 \cos \frac{x}{2} + C \right) \]

### Step 5: Final Result
This simplifies to:
\[ I = -2\sqrt{2} \cos \frac{x}{2} + C \]

### Final Answer
\[ \int \frac{\sin x}{\sqrt{1 + \cos x}} \, dx = -2\sqrt{2} \cos \frac{x}{2} + C \]

---