The value of `I=int(dx)/(2xsqrt(1-x)sqrt((2-x)+sqrt(1-x)))=-1/2{log(z+3/2+sqrt(z^(2)+3z+3))}+1/2|log s-1/2+sqrt(s^(2)-s+1)|+C` and `s-z=k/x`, then value of k, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( k \) in the given integral equation, we will follow the steps outlined in the video transcript to derive the solution systematically.

### Step 1: Set up the integral
We start with the integral given in the question:

\[ I = \int \frac{dx}{2x \sqrt{1-x} \sqrt{(2-x) + \sqrt{1-x}}} \]

### Step 2: Substitute
We will use the substitution \( 1 - x = t^2 \). Thus, \( dx = -2t \, dt \). This transforms our integral:

\[ I = -\int \frac{2t \, dt}{2 \sqrt{1-t^2} \sqrt{(2 - (1 - t^2)) + t}} = -\int \frac{t \, dt}{\sqrt{1 - t^2} \sqrt{(1 + t^2) + t}} \]

### Step 3: Simplify the integral
Now we simplify the expression inside the integral. The integral becomes:

\[ I = -\int \frac{dt}{\sqrt{1 - t^2} \sqrt{1 + t^2 + t}} \]

### Step 4: Partial fraction decomposition
Next, we will break the integral into partial fractions. We can express the integrand as:

\[ \frac{1}{(1 + t)(1 - \sqrt{t^2 + t - 1})} \]

### Step 5: Rewrite the integral
We can rewrite the integral \( I \) as:

\[ I = \frac{1}{2} I_1 - \frac{1}{2} I_2 \]

Where \( I_1 \) and \( I_2 \) are the integrals we need to evaluate.

### Step 6: Evaluate \( I_1 \)
For \( I_1 \):

\[ I_1 = \int \frac{dt}{t - 1} + \int \frac{dt}{t + 1 \sqrt{t^2 + t + 1}} \]

Using the substitution \( t - 1 = \frac{1}{z} \) for the first integral, we can evaluate it.

### Step 7: Evaluate \( I_2 \)
For \( I_2 \):

Using the substitution \( t + 1 = \frac{1}{s} \), we can evaluate this integral similarly.

### Step 8: Combine results
After evaluating both integrals, we will have:

\[ I = -\frac{1}{2} \left( \log(z + \frac{3}{2} + \sqrt{z^2 + 3z + 3}) - \log(s - \frac{1}{2} + \sqrt{s^2 - s + 1}) \right) + C \]

### Step 9: Relate \( s \) and \( z \)
Given \( s - z = \frac{k}{x} \), we can express \( s \) in terms of \( z \):

\[ s = z + \frac{k}{x} \]

### Step 10: Compare coefficients
By comparing the coefficients from the integrals and the transformations, we find that:

\[ \frac{2}{x} = \frac{k}{x} \]

### Step 11: Solve for \( k \)
Thus, we conclude that:

\[ k = 2 \]

### Final Answer
The value of \( k \) is \( \boxed{2} \).

---