Choose the correct Answer of the Following Questions : `int_0^(pi/2) (sinx dx)/sqrt(1 + cosx)` is equal to

Options

A. `2(sqrt2 + 1)`
B. `2(sqrt2 - 1)`
C. `1- sqrt2`
D. 0

Answer

To solve the integral \( I = \int_0^{\frac{\pi}{2}} \frac{\sin x}{\sqrt{1 + \cos x}} \, dx \), we will follow these steps:

### Step 1: Simplify the integrand
We start with the integrand \( \frac{\sin x}{\sqrt{1 + \cos x}} \). We can use trigonometric identities to simplify this expression.

Recall that:
- \( \sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} \)
- \( 1 + \cos x = 2 \cos^2 \frac{x}{2} \)

Thus, we can rewrite the integrand:
\[ \frac{\sin x}{\sqrt{1 + \cos x}} = \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{\sqrt{2 \cos^2 \frac{x}{2}}} \]
This simplifies to:
\[ \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{\sqrt{2} \cos \frac{x}{2}} = \frac{2 \sin \frac{x}{2}}{\sqrt{2}} = \sqrt{2} \sin \frac{x}{2} \]

### Step 2: Rewrite the integral
Now we can rewrite the integral in terms of the simplified integrand:
\[ I = \int_0^{\frac{\pi}{2}} \sqrt{2} \sin \frac{x}{2} \, dx \]

### Step 3: Factor out the constant
Since \( \sqrt{2} \) is a constant, we can factor it out of the integral:
\[ I = \sqrt{2} \int_0^{\frac{\pi}{2}} \sin \frac{x}{2} \, dx \]

### Step 4: Perform the integration
To integrate \( \sin \frac{x}{2} \), we use the formula for the integral of sine:
\[ \int \sin mx \, dx = -\frac{1}{m} \cos mx + C \]
Here, \( m = \frac{1}{2} \), so:
\[ \int \sin \frac{x}{2} \, dx = -2 \cos \frac{x}{2} \]

Now we evaluate the definite integral:
\[ \int_0^{\frac{\pi}{2}} \sin \frac{x}{2} \, dx = \left[-2 \cos \frac{x}{2}\right]_0^{\frac{\pi}{2}} = -2 \left( \cos \frac{\pi}{4} - \cos 0 \right) \]
Calculating the values:
\[ \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}, \quad \cos 0 = 1 \]
Thus:
\[ -2 \left( \frac{1}{\sqrt{2}} - 1 \right) = -2 \left( \frac{1 - \sqrt{2}}{\sqrt{2}} \right) = \frac{2(\sqrt{2} - 1)}{\sqrt{2}} \]

### Step 5: Combine results
Now substituting back into the integral:
\[ I = \sqrt{2} \cdot \frac{2(\sqrt{2} - 1)}{\sqrt{2}} = 2(\sqrt{2} - 1) \]

### Final Answer
Thus, the value of the integral is:
\[ I = 2(\sqrt{2} - 1) \]