`int (2+cot^(2) x) dx`

Answer

To solve the integral \( \int (2 + \cot^2 x) \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start by rewriting the integral:
\[ \int (2 + \cot^2 x) \, dx = \int 2 \, dx + \int \cot^2 x \, dx \]

### Step 2: Integrate the first part
The integral of \( 2 \) with respect to \( x \) is straightforward:
\[ \int 2 \, dx = 2x \]

### Step 3: Integrate the second part
Next, we need to integrate \( \cot^2 x \). We can use the identity:
\[ \cot^2 x = \csc^2 x - 1 \]
Thus, we can rewrite the integral:
\[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx \]
This can be split into two separate integrals:
\[ \int \cot^2 x \, dx = \int \csc^2 x \, dx - \int 1 \, dx \]

### Step 4: Integrate \( \csc^2 x \)
The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x \]

### Step 5: Integrate the constant
The integral of \( 1 \) is:
\[ \int 1 \, dx = x \]

### Step 6: Combine the results
Now, we can combine all parts:
\[ \int \cot^2 x \, dx = -\cot x - x \]

### Step 7: Write the final result
Putting it all together, we have:
\[ \int (2 + \cot^2 x) \, dx = 2x + (-\cot x - x) + C \]
This simplifies to:
\[ \int (2 + \cot^2 x) \, dx = x - \cot x + C \]

### Final Answer:
\[ \int (2 + \cot^2 x) \, dx = x - \cot x + C \]