`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 expression under the square root
We start with the integral:
\[ \int \frac{1}{\sqrt{1 + x - x^2}} \, dx \]
First, we rewrite the expression inside the square root:
\[ 1 + x - x^2 = 1 - (x^2 - x) = 1 - (x^2 - x + \frac{1}{4} - \frac{1}{4}) = 1 - \left( (x - \frac{1}{2})^2 - \frac{1}{4} \right) \]
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:
\[ \int \frac{1}{\sqrt{\frac{5}{4} - (x - \frac{1}{2})^2}} \, dx \]
This can be simplified further:
\[ = \int \frac{1}{\sqrt{\left(\frac{\sqrt{5}}{2}\right)^2 - (x - \frac{1}{2})^2}} \, dx \]

### Step 3: Use trigonometric substitution
We can use the substitution:
\[ x - \frac{1}{2} = \frac{\sqrt{5}}{2} \sin \theta \]
Then, \( dx = \frac{\sqrt{5}}{2} \cos \theta \, d\theta \).

### Step 4: Change the limits of integration
The integral becomes:
\[ = \int \frac{\frac{\sqrt{5}}{2} \cos \theta}{\sqrt{\left(\frac{\sqrt{5}}{2}\right)^2 - \left(\frac{\sqrt{5}}{2} \sin \theta\right)^2}} \, d\theta \]
This simplifies to:
\[ = \int \frac{\frac{\sqrt{5}}{2} \cos \theta}{\sqrt{\frac{5}{4}(1 - \sin^2 \theta)}} \, d\theta \]
\[ = \int \frac{\frac{\sqrt{5}}{2} \cos \theta}{\frac{\sqrt{5}}{2} \cos \theta} \, d\theta = \int d\theta \]

### Step 5: Integrate
The integral of \( d\theta \) is simply:
\[ \theta + C \]

### Step 6: Back substitute
Now we need to back substitute for \( \theta \):
\[ \theta = \sin^{-1}\left(\frac{2(x - \frac{1}{2})}{\sqrt{5}}\right) = \sin^{-1}\left(\frac{2x - 1}{\sqrt{5}}\right) \]

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