`int(dx)/(sqrt(x-x^(2)))` equal is :

Options

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

Answer

To solve the integral \( I = \int \frac{dx}{\sqrt{x - x^2}} \), we can follow these steps:

### Step 1: Rewrite the integrand
First, we can rewrite the expression under the square root:
\[ x - x^2 = x(1 - x) \]
Thus, we have:
\[ I = \int \frac{dx}{\sqrt{x(1 - x)}} \]

### Step 2: Simplify the expression
Next, we can factor out \( \frac{1}{4} \) from the expression under the square root:
\[ \sqrt{x(1 - x)} = \sqrt{\frac{1}{4} - (x - \frac{1}{2})^2} \]
This leads us to rewrite the integral as:
\[ I = \int \frac{dx}{\sqrt{\frac{1}{4} - (x - \frac{1}{2})^2}} \]

### Step 3: Use substitution
Now, we can use the substitution:
\[ u = x - \frac{1}{2} \quad \Rightarrow \quad dx = du \]
Then the limits change accordingly, and we have:
\[ I = \int \frac{du}{\sqrt{\frac{1}{4} - u^2}} \]

### Step 4: Apply the standard integral formula
We recognize that this integral matches the standard form:
\[ \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1} \left( \frac{u}{a} \right) + C \]
where \( a = \frac{1}{2} \). Thus, we can write:
\[ I = \sin^{-1} \left( \frac{u}{\frac{1}{2}} \right) + C \]

### Step 5: Substitute back for \( u \)
Substituting back \( u = x - \frac{1}{2} \):
\[ I = \sin^{-1} \left( 2\left(x - \frac{1}{2}\right) \right) + C \]
This simplifies to:
\[ I = \sin^{-1}(2x - 1) + C \]

### Final Answer
Thus, the final result for the integral is:
\[ I = \sin^{-1}(2x - 1) + C \]