`int(1)/((1+sqrtx)sqrt(x-x^(2)))dx` is equal to

Options

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

Explanation

To solve the integral \( \int \frac{1}{(1+\sqrt{x})\sqrt{x-x^2}} \, dx \), we will perform a substitution and simplify the expression step by step.

### Step 1: Substitution
Let \( \sqrt{x} = \sin(t) \). Then, we have:
\[ x = \sin^2(t) \]
Differentiating both sides gives:
\[ dx = 2\sin(t)\cos(t) \, dt \]

### Step 2: Substitute in the Integral
Now, we substitute \( \sqrt{x} \) and \( dx \) into the integral:
\[ \sqrt{x - x^2} = \sqrt{\sin^2(t) - \sin^4(t)} = \sqrt{\sin^2(t)(1 - \sin^2(t))} = \sqrt{\sin^2(t) \cos^2(t)} = \sin(t) \cos(t) \]
Thus, the integral becomes:
\[ \int \frac{1}{(1+\sin(t)) \sin(t) \cos(t)} \cdot 2\sin(t)\cos(t) \, dt \]

### Step 3: Simplify the Integral
The \( \sin(t) \cos(t) \) terms cancel out:
\[ = \int \frac{2}{1+\sin(t)} \, dt \]

### Step 4: Further Simplification
To simplify \( \frac{2}{1+\sin(t)} \), we multiply the numerator and denominator by \( 1 - \sin(t) \):
\[ = \int \frac{2(1 - \sin(t))}{(1+\sin(t))(1-\sin(t))} \, dt = \int \frac{2(1 - \sin(t))}{1 - \sin^2(t)} \, dt = \int \frac{2(1 - \sin(t))}{\cos^2(t)} \, dt \]
This can be separated into two integrals:
\[ = 2\int \sec^2(t) \, dt - 2\int \frac{\sin(t)}{\cos^2(t)} \, dt \]

### Step 5: Integrate
The integrals can be solved as follows:
1. \( \int \sec^2(t) \, dt = \tan(t) + C \)
2. \( \int \frac{\sin(t)}{\cos^2(t)} \, dt = -\frac{1}{\cos(t)} + C \)

Thus, we have:
\[ = 2\tan(t) + 2\sec(t) + C \]

### Step 6: Back Substitute
Recall that:
- \( \tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{\sqrt{x}}{\sqrt{1-x}} \)
- \( \sec(t) = \frac{1}{\cos(t)} = \frac{1}{\sqrt{1-x}} \)

Putting these back into our expression gives:
\[ = 2\left(\frac{\sqrt{x}}{\sqrt{1-x}}\right) + 2\left(\frac{1}{\sqrt{1-x}}\right) + C \]
Combining these terms:
\[ = \frac{2\sqrt{x} + 2}{\sqrt{1-x}} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{1}{(1+\sqrt{x})\sqrt{x-x^2}} \, dx = \frac{2\sqrt{x} + 2}{\sqrt{1-x}} + C \]