Evaluate the following :
`int (x-1)/(x+1) dx.`

Answer

To evaluate the integral \(\int \frac{x-1}{x+1} \, dx\), we can follow these steps:

### Step 1: Rewrite the integrand
We can rewrite the integrand by adding and subtracting 1:
\[ \frac{x-1}{x+1} = \frac{x+1-2}{x+1} = \frac{x+1}{x+1} - \frac{2}{x+1} \]
This simplifies to:
\[ \frac{x-1}{x+1} = 1 - \frac{2}{x+1} \]

### Step 2: Split the integral
Now we can split the integral into two parts:
\[ \int \frac{x-1}{x+1} \, dx = \int \left(1 - \frac{2}{x+1}\right) \, dx \]

### Step 3: Integrate each term
Now we can integrate each term separately:
1. The integral of \(1\) is \(x\).
2. The integral of \(-\frac{2}{x+1}\) is \(-2 \ln |x+1|\).

Putting it all together, we have:
\[ \int \frac{x-1}{x+1} \, dx = x - 2 \ln |x+1| + C \]
where \(C\) is the constant of integration.

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

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