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

Answer

To evaluate the integral \( I = \int \frac{x-1}{\sqrt{x^2 - x}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{x-1}{\sqrt{x^2 - x}} \, dx \]

### Step 2: Simplify the Numerator
We can express \( x - 1 \) in a different form. Notice that:
\[ x - 1 = \frac{2x - 2}{2} = \frac{2(x - 1)}{2} = \frac{2x - 1 - 1}{2} \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{1}{2} \left( \frac{2x - 1}{\sqrt{x^2 - x}} - \frac{1}{\sqrt{x^2 - x}} \right) \, dx \]

### Step 3: Split the Integral
Now we can split the integral into two parts:
\[ I = \frac{1}{2} \int \frac{2x - 1}{\sqrt{x^2 - x}} \, dx - \frac{1}{2} \int \frac{1}{\sqrt{x^2 - x}} \, dx \]

### Step 4: Substitution for the First Integral
Let \( y = x^2 - x \). Then, the derivative \( dy = (2x - 1) \, dx \). Thus, we can rewrite the first integral:
\[ \int \frac{2x - 1}{\sqrt{x^2 - x}} \, dx = \int \frac{1}{\sqrt{y}} \, dy \]
This evaluates to:
\[ 2\sqrt{y} + C_1 = 2\sqrt{x^2 - x} + C_1 \]

### Step 5: Evaluate the Second Integral
For the second integral, we have:
\[ \int \frac{1}{\sqrt{x^2 - x}} \, dx \]
This integral can be simplified further. We rewrite \( x^2 - x \) as:
\[ x^2 - x = \left( x - \frac{1}{2} \right)^2 - \frac{1}{4} \]
Thus, we can use the formula for the integral of the form \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \):
\[ \int \frac{1}{\sqrt{\left( x - \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2}} \, dx = \ln \left| x - \frac{1}{2} + \sqrt{\left( x - \frac{1}{2} \right)^2 - \frac{1}{4}} \right| + C_2 \]

### Step 6: Combine the Results
Combining both parts, we get:
\[ I = \frac{1}{2} \left( 2\sqrt{x^2 - x} \right) - \frac{1}{2} \ln \left| x - \frac{1}{2} + \sqrt{\left( x - \frac{1}{2} \right)^2 - \frac{1}{4}} \right| + C \]
This simplifies to:
\[ I = \sqrt{x^2 - x} - \frac{1}{2} \ln \left| x - \frac{1}{2} + \sqrt{\left( x - \frac{1}{2} \right)^2 - \frac{1}{4}} \right| + C \]

### Final Answer
Thus, the evaluated integral is:
\[ I = \sqrt{x^2 - x} - \frac{1}{2} \ln \left| x - \frac{1}{2} + \sqrt{\left( x - \frac{1}{2} \right)^2 - \frac{1}{4}} \right| + C \]