Evaluate `int (3x-1)/((x-2)^(2))dx`

Answer

To evaluate the integral \( \int \frac{3x - 1}{(x - 2)^2} \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
We start with the integral:
\[ I = \int \frac{3x - 1}{(x - 2)^2} \, dx \]
We can rewrite the numerator \( 3x - 1 \) as \( 3(x - 2) + 6 - 1 \):
\[ 3x - 1 = 3(x - 2) + 6 - 1 = 3(x - 2) + 5 \]
Thus, we can express the integral as:
\[ I = \int \frac{3(x - 2) + 5}{(x - 2)^2} \, dx \]

### Step 2: Split the integral
Now we can split the integral into two parts:
\[ I = \int \frac{3(x - 2)}{(x - 2)^2} \, dx + \int \frac{5}{(x - 2)^2} \, dx \]
This simplifies to:
\[ I = \int \frac{3}{x - 2} \, dx + 5 \int (x - 2)^{-2} \, dx \]

### Step 3: Evaluate the first integral
The first integral is:
\[ \int \frac{3}{x - 2} \, dx = 3 \ln |x - 2| \]

### Step 4: Evaluate the second integral
The second integral can be evaluated using the power rule:
\[ \int (x - 2)^{-2} \, dx = \frac{(x - 2)^{-1}}{-1} = -\frac{1}{x - 2} \]
Thus,
\[ 5 \int (x - 2)^{-2} \, dx = -\frac{5}{x - 2} \]

### Step 5: Combine the results
Combining both parts, we have:
\[ I = 3 \ln |x - 2| - \frac{5}{x - 2} + C \]
where \( C \) is the constant of integration.

### Final Answer
The evaluated integral is:
\[ \int \frac{3x - 1}{(x - 2)^2} \, dx = 3 \ln |x - 2| - \frac{5}{x - 2} + C \]