`int((3x-2)^2)/x^2dx`=?

Answer

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

### Step 1: Expand the numerator
We start by expanding the expression \((3x - 2)^2\).
\[ (3x - 2)^2 = 9x^2 - 12x + 4 \]
So, we rewrite the integral:
\[ \int \frac{(3x-2)^2}{x^2} \, dx = \int \frac{9x^2 - 12x + 4}{x^2} \, dx \]

### Step 2: Separate the terms in the integral
Next, we can separate the integral into three simpler integrals:
\[ \int \frac{9x^2}{x^2} \, dx - \int \frac{12x}{x^2} \, dx + \int \frac{4}{x^2} \, dx \]
This simplifies to:
\[ \int 9 \, dx - \int \frac{12}{x} \, dx + \int 4x^{-2} \, dx \]

### Step 3: Integrate each term
Now we can integrate each term separately:
1. \(\int 9 \, dx = 9x\)
2. \(\int \frac{12}{x} \, dx = 12 \ln |x|\)
3. \(\int 4x^{-2} \, dx = 4 \cdot \left(-\frac{1}{x}\right) = -\frac{4}{x}\)

### Step 4: Combine the results
Now we combine all the results of the integrals:
\[ 9x - 12 \ln |x| - \frac{4}{x} + C \]
where \(C\) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{(3x-2)^2}{x^2} \, dx = 9x - 12 \ln |x| - \frac{4}{x} + C \]