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

Options

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

Explanation

To solve the integral

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

we will follow these steps:

### Step 1: Simplify the Integral
We start by rewriting the integral:

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

### Step 2: Factor Out Common Terms
We can factor \(x\) out from the square roots in the denominator:

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

### Step 3: Rewrite the Numerator
Next, we take \(x^2\) common from the numerator:

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

### Step 4: Cancel Out \(x^2\)
The \(x^2\) in the numerator and denominator cancels out:

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

### Step 5: Substitution
Now, we will apply substitution. Let

\[ t = \sqrt{x + 2 - \frac{1}{x}}. \]

Differentiating both sides gives:

\[ dt = \left(\frac{1}{2\sqrt{x + 2 - \frac{1}{x}}}\right) \left(1 + \frac{1}{x^2}\right) dx. \]

### Step 6: Express \(dx\) in Terms of \(dt\)
From the above, we can express \(dx\):

\[ dx = 2t \cdot dt \cdot \frac{1}{1 + \frac{1}{x^2}}. \]

### Step 7: Substitute Back into the Integral
Substituting \(dx\) back into the integral gives:

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

### Step 8: Solve the Integral
The integral now resembles the standard form:

\[ \int \frac{dt}{\sqrt{1 - t^2}} = \sin^{-1}(t) + C. \]

Thus,

\[ I = 2 \sin^{-1}(t) + C. \]

### Step 9: Resubstitute \(t\)
Now, we substitute back for \(t\):

\[ I = 2 \sin^{-1}\left(\sqrt{x + 2 - \frac{1}{x}}\right) + C. \]

### Final Answer
Therefore, the solution to the integral is:

\[ I = 2 \sin^{-1}\left(\sqrt{x + 2 - \frac{1}{x}}\right) + C. \]