`int(3cotx - 2 tanx)^(2) dx`

Explanation

To solve the integral \( \int (3 \cot x - 2 \tan x)^2 \, dx \), we will follow these steps:

### Step 1: Expand the integrand
We start by expanding the expression \( (3 \cot x - 2 \tan x)^2 \):
\[ (3 \cot x - 2 \tan x)^2 = (3 \cot x)^2 - 2 \cdot (3 \cot x)(2 \tan x) + (2 \tan x)^2 \]
This simplifies to:
\[ 9 \cot^2 x - 12 \cot x \tan x + 4 \tan^2 x \]

### Step 2: Rewrite the integral
Now we can rewrite the integral:
\[ \int (3 \cot x - 2 \tan x)^2 \, dx = \int (9 \cot^2 x - 12 \cot x \tan x + 4 \tan^2 x) \, dx \]

### Step 3: Use trigonometric identities
We can use the identities \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \) and \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \). Also, we know that \( \cot x \tan x = 1 \). Thus, we can rewrite the integral:
\[ = \int \left( 9 \cot^2 x + 4 \tan^2 x - 12 \right) \, dx \]

### Step 4: Substitute the identities
Using the identity \( \cot^2 x = \csc^2 x - 1 \) and \( \tan^2 x = \sec^2 x - 1 \):
\[ = \int \left( 9 (\csc^2 x - 1) + 4 (\sec^2 x - 1) - 12 \right) \, dx \]
This simplifies to:
\[ = \int \left( 9 \csc^2 x + 4 \sec^2 x - 9 - 4 - 12 \right) \, dx \]
\[ = \int \left( 9 \csc^2 x + 4 \sec^2 x - 25 \right) \, dx \]

### Step 5: Integrate each term
Now we can integrate each term separately:
\[ = 9 \int \csc^2 x \, dx + 4 \int \sec^2 x \, dx - 25 \int dx \]
The integrals are:
- \( \int \csc^2 x \, dx = -\cot x \)
- \( \int \sec^2 x \, dx = \tan x \)
- \( \int dx = x \)

So we have:
\[ = 9(-\cot x) + 4\tan x - 25x + C \]

### Final Answer
Thus, the final result is:
\[ -9 \cot x + 4 \tan x - 25x + C \]