The integral `int(1)/((1+sqrt(x))sqrt(x-x^(2)))dx` is equal to (where C is the constant of integration)

Options

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

Explanation

To solve the integral

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

we can start by simplifying the expression under the integral.

### Step 1: Rewrite the integral
Notice that \( \sqrt{x - x^2} = \sqrt{x(1 - x)} \). Thus, we can rewrite the integral as:

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

### Step 2: Use a substitution
Let's use the substitution \( x = \sin^2(\theta) \). Then, we have:

\[ dx = 2 \sin(\theta) \cos(\theta) \, d\theta = \sin(2\theta) \, d\theta. \]

Now, substituting \( x = \sin^2(\theta) \) gives:

\[ \sqrt{x} = \sin(\theta) \quad \text{and} \quad 1 - x = \cos^2(\theta). \]

Thus,

\[ \sqrt{x(1 - x)} = \sqrt{\sin^2(\theta) \cos^2(\theta)} = \sin(\theta) \cos(\theta). \]

### Step 3: Substitute into the integral
Now substituting everything into the integral:

\[ I = \int \frac{\sin(2\theta)}{(1 + \sin(\theta)) \sin(\theta) \cos(\theta)} \, d\theta. \]

### Step 4: Simplify the integral
We can simplify the integral further:

\[ I = \int \frac{2 \sin(\theta) \cos(\theta)}{(1 + \sin(\theta)) \sin(\theta) \cos(\theta)} \, d\theta = \int \frac{2}{1 + \sin(\theta)} \, d\theta. \]

### Step 5: Use another substitution
To solve this integral, we can use the substitution \( u = 1 + \sin(\theta) \), then \( du = \cos(\theta) d\theta \). The integral becomes:

\[ I = 2 \int \frac{1}{u} \, du = 2 \ln |u| + C = 2 \ln |1 + \sin(\theta)| + C. \]

### Step 6: Back substitute
Now we need to convert back to \( x \). Recall that \( \sin(\theta) = \sqrt{x} \):

\[ I = 2 \ln |1 + \sqrt{x}| + C. \]

### Final Answer
Thus, the integral

\[ \int \frac{1}{(1 + \sqrt{x}) \sqrt{x - x^2}} \, dx = 2 \ln |1 + \sqrt{x}| + C. \]