`int cot^(2) x dx = ?`

Options

A. `-cot x -x + C`
B. `cot x-x + C`
C. `-cotx + x + C`
D. `cot x + x + C`

Explanation

To solve the integral \( \int \cot^2 x \, dx \), we can use the identity relating cotangent to cosecant. Here’s a step-by-step solution:

### Step 1: Use the identity for cotangent
Recall that:
\[ \cot^2 x = \csc^2 x - 1 \]
This allows us to rewrite the integral:
\[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx \]

### Step 2: Split the integral
Now we can split the integral into two parts:
\[ \int \cot^2 x \, dx = \int \csc^2 x \, dx - \int 1 \, dx \]

### Step 3: Integrate each term
1. The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x + C_1 \]
2. The integral of \( 1 \) is:
\[ \int 1 \, dx = x + C_2 \]

### Step 4: Combine the results
Putting it all together:
\[ \int \cot^2 x \, dx = (-\cot x + C_1) - (x + C_2) \]
Simplifying this gives:
\[ \int \cot^2 x \, dx = -\cot x - x + C \]
where \( C = C_1 - C_2 \) is the constant of integration.

### Final Answer
Thus, the integral is:
\[ \int \cot^2 x \, dx = -\cot x - x + C \]

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