If `int sqrt2 sqrt (1 + sin x ) dx =- 4 cos (ax +b)+C` then the value of (a,b) is :

Options

A. `((1)/(2) , (pi)/(4))`
B. `(1, (pi)/(2))`
C. `(1,1)`
D. None of these

Answer

To solve the integral \( \int \sqrt{2} \sqrt{1 + \sin x} \, dx = -4 \cos(ax + b) + C \) and find the values of \( (a, b) \), we will follow these steps:

### Step 1: Simplify the Integral
We start with the integral:
\[ I = \int \sqrt{2} \sqrt{1 + \sin x} \, dx \]
We can factor out \( \sqrt{2} \):
\[ I = \sqrt{2} \int \sqrt{1 + \sin x} \, dx \]

### Step 2: Rewrite the Expression
Next, we rewrite \( 1 + \sin x \) using the identity:
\[ 1 + \sin x = \frac{(1 + \sin x)(1 - \sin x)}{1 - \sin x} = \frac{1 - \sin^2 x}{1 - \sin x} = \frac{\cos^2 x}{1 - \sin x} \]
Thus,
\[ \sqrt{1 + \sin x} = \sqrt{\frac{\cos^2 x}{1 - \sin x}} = \frac{\cos x}{\sqrt{1 - \sin x}} \]
So the integral becomes:
\[ I = \sqrt{2} \int \frac{\cos x}{\sqrt{1 - \sin x}} \, dx \]

### Step 3: Substitution
Let \( t = 1 - \sin x \), then \( dt = -\cos x \, dx \) or \( \cos x \, dx = -dt \). The integral now transforms to:
\[ I = -\sqrt{2} \int \frac{1}{\sqrt{t}} \, dt = -\sqrt{2} \cdot 2\sqrt{t} + C = -2\sqrt{2\sqrt{1 - \sin x}} + C \]

### Step 4: Express in Terms of \( \sin \) and \( \cos \)
Now we rewrite \( \sqrt{1 - \sin x} \):
\[ \sqrt{1 - \sin x} = \sqrt{\cos^2\left(\frac{x}{2}\right)} = \cos\left(\frac{x}{2}\right) \]
Thus,
\[ I = -2\sqrt{2} \cos\left(\frac{x}{2}\right) + C \]

### Step 5: Compare with Given Expression
We need to compare this with the expression:
\[ -4 \cos(ax + b) + C \]
From our integral, we have:
\[ -2\sqrt{2} \cos\left(\frac{x}{2}\right) = -4 \cos(ax + b) \]
This implies:
\[ \sqrt{2} \cos\left(\frac{x}{2}\right) = 2 \cos(ax + b) \]

### Step 6: Identify \( a \) and \( b \)
From the comparison, we can deduce:
1. The coefficient \( a \) must be \( \frac{1}{2} \) since \( \frac{x}{2} \) can be expressed as \( ax \) where \( a = \frac{1}{2} \).
2. The phase shift \( b \) corresponds to \( 0 \) since there is no additional term in the cosine.

Thus, we find:
\[ (a, b) = \left(\frac{1}{2}, 0\right) \]

### Final Answer
The values of \( (a, b) \) are:
\[ \left(\frac{1}{2}, 0\right) \]