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

Answer

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

### Step 1: Identify the substitution
We notice that the derivative of the expression inside the square root, \( x^2 - x - 1 \), is \( 2x - 1 \). This suggests that we can use substitution to simplify the integral.

**Hint:** Look for a function whose derivative appears in the numerator of the integral.

### Step 2: Make the substitution
Let:
\[ t = x^2 - x - 1 \]
Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 2x - 1 \quad \Rightarrow \quad dt = (2x - 1) \, dx \]
This means we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{2x - 1} \]

**Hint:** When making a substitution, always differentiate to find \( dt \).

### Step 3: Rewrite the integral
Substituting \( t \) and \( dx \) into the integral gives:
\[ \int \frac{2x - 1}{\sqrt{t}} \cdot \frac{dt}{2x - 1} = \int \frac{dt}{\sqrt{t}} \]

**Hint:** Simplify the integral after substitution to make it easier to integrate.

### Step 4: Integrate
The integral \( \int \frac{dt}{\sqrt{t}} \) can be rewritten as:
\[ \int t^{-1/2} \, dt \]
Using the power rule for integration, we have:
\[ \int t^{-1/2} \, dt = 2t^{1/2} + C \]

**Hint:** Remember the power rule for integration: \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \) for \( n \neq -1 \).

### Step 5: Substitute back
Now, we substitute back \( t = x^2 - x - 1 \):
\[ 2\sqrt{t} + C = 2\sqrt{x^2 - x - 1} + C \]

**Hint:** Always revert back to the original variable after integrating.

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{2x-1}{\sqrt{x^2 - x - 1}} \, dx = 2\sqrt{x^2 - x - 1} + C \]