`int(sqrt(1-x^(2))-x)/(sqrt(1-x^(2))(1+xsqrt(1-x^(2))))dx` is

Options

A. `2tan^(-1)(x+sqrt(1-x^(2)))+c`
B. `tan^(-1)(x+sqrt(1-x^(2)))+c`
C. `2tan^(-1)(x-sqrt(1-x^(2)))+c`
D. `2cot^(-1)(x+sqrt(1-x^(2)))+c`

Explanation

To solve the integral

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

we can use the substitution \( x = \sin \theta \).

### Step-by-Step Solution:

1. **Substitution**: Let \( x = \sin \theta \). Then, \( dx = \cos \theta \, d\theta \).

2. **Transform the Integral**:
- We know that \( \sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta \).
- Substitute these into the integral:
\[ \int \frac{\cos \theta - \sin \theta}{\cos \theta (1 + \sin \theta \cos \theta)} \cos \theta \, d\theta. \]
- This simplifies to:
\[ \int \frac{\cos^2 \theta - \sin \theta \cos \theta}{\cos^2 \theta (1 + \sin \theta \cos \theta)} \, d\theta. \]

3. **Simplifying the Denominator**:
- The denominator becomes:
\[ 1 + \sin \theta \cos \theta = 1 + \frac{1}{2} \sin(2\theta). \]
- To make it easier, multiply the numerator and denominator by 2:
\[ \int \frac{2(\cos^2 \theta - \sin \theta \cos \theta)}{2\cos^2 \theta + \sin(2\theta)} \, d\theta. \]

4. **Rearranging**:
- The integral can be rewritten as:
\[ 2 \int \frac{\cos \theta - \sin \theta}{1 + \sin \theta + \cos \theta} \, d\theta. \]

5. **Substitution for Integration**:
- Let \( t = \sin \theta + \cos \theta \). Then, \( dt = (\cos \theta - \sin \theta) \, d\theta \).
- The integral now becomes:
\[ 2 \int \frac{dt}{1 + t^2}. \]

6. **Integral of the Result**:
- The integral of \( \frac{1}{1 + t^2} \) is \( \tan^{-1}(t) \):
\[ 2 \tan^{-1}(t) + C. \]

7. **Back Substitution**:
- Substitute back \( t = \sin \theta + \cos \theta \):
\[ 2 \tan^{-1}(\sin \theta + \cos \theta) + C. \]
- Since \( \sin \theta = x \) and \( \cos \theta = \sqrt{1 - x^2} \), we have:
\[ 2 \tan^{-1}(x + \sqrt{1 - x^2}) + C. \]

### Final Answer:
Thus, the value of the integral is:

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