https://file.aiquickdraw.com/mathbot/file/1727346330616qiwveaqe.jpg

The expression in the image is:

\[
\int \cot^2(x) \, dx
\]

To solve this integral, we can proceed by using trigonometric identities. Here's the detailed process:

### Step 1: Rewrite \(\cot^2(x)\) using a trigonometric identity
We know that:
\[
\cot^2(x) = \csc^2(x) - 1
\]
This identity comes from the Pythagorean identity for cotangent:
\[
\csc^2(x) = 1 + \cot^2(x)
\]
So the integral becomes:
\[
\int \cot^2(x) \, dx = \int (\csc^2(x) - 1) \, dx
\]

### Step 2: Break the integral into two simpler integrals
\[
\int (\csc^2(x) - 1) \, dx = \int \csc^2(x) \, dx - \int 1 \, dx
\]

### Step 3: Integrate both terms
1. The integral of \(\csc^2(x)\) is \(-\cot(x)\):
   \[
   \int \csc^2(x) \, dx = -\cot(x)
   \]
2. The integral of \(1\) is just \(x\):
   \[
   \int 1 \, dx = x
   \]

### Step 4: Combine the results
So, the integral becomes:
\[
-\cot(x) - x + C
\]
where \(C\) is the constant of integration.

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

Let me know if you'd like any further details or clarification!

### Related Questions:
1. What are the other Pythagorean identities used in trigonometry?
2. How do you integrate \(\csc(x)\) and \(\sec(x)\) functions?
3. Can the integral of \(\cot^2(x)\) be solved using substitution methods?
4. What is the integral of \(\sec^2(x)\)?
5. How do we derive the identity \(\cot^2(x) = \csc^2(x) - 1\)?

### Tip:
Memorizing basic trigonometric integrals like \(\int \csc^2(x) dx\) and \(\int \sec^2(x) dx\) can significantly speed up solving trigonometric integrals!

Learn how to evaluate the integral of cot^2(x) using trigonometric identities and integration techniques. This detailed guide covers the step-by-step process of solving ∫ cot^2(x) dx and highlights key formulas and theorems. Suitable for college-level calculus students.

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