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

Answer

To solve the integral \( \int \frac{1}{\sqrt{1 + x - x^2}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Denominator
We start with the expression in the denominator:
\[ 1 + x - x^2 \]
We can rearrange this as:
\[ 1 - (x^2 - x) = 1 - (x^2 - x + \frac{1}{4} - \frac{1}{4}) = 1 - \left( (x - \frac{1}{2})^2 - \frac{1}{4} \right) \]
This simplifies to:
\[ 1 - (x - \frac{1}{2})^2 + \frac{1}{4} = \frac{5}{4} - (x - \frac{1}{2})^2 \]

### Step 2: Substitute into the Integral
Now we can rewrite the integral:
\[ \int \frac{1}{\sqrt{1 + x - x^2}} \, dx = \int \frac{1}{\sqrt{\frac{5}{4} - (x - \frac{1}{2})^2}} \, dx \]

### Step 3: Apply the Formula for the Integral
We can now use the formula for the integral:
\[ \int \frac{1}{\sqrt{a^2 - u^2}} \, du = \sin^{-1} \left( \frac{u}{a} \right) + C \]
In our case, \( a = \frac{\sqrt{5}}{2} \) and \( u = x - \frac{1}{2} \).

### Step 4: Substitute Values into the Formula
Thus, we have:
\[ \int \frac{1}{\sqrt{\frac{5}{4} - (x - \frac{1}{2})^2}} \, dx = \sin^{-1} \left( \frac{x - \frac{1}{2}}{\frac{\sqrt{5}}{2}} \right) + C \]

### Step 5: Simplify the Expression
We can simplify the argument of the sine inverse:
\[ \sin^{-1} \left( \frac{2(x - \frac{1}{2})}{\sqrt{5}} \right) + C = \sin^{-1} \left( \frac{2x - 1}{\sqrt{5}} \right) + C \]

### Final Answer
Thus, the final result for the integral is:
\[ \int \frac{1}{\sqrt{1 + x - x^2}} \, dx = \sin^{-1} \left( \frac{2x - 1}{\sqrt{5}} \right) + C \]

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