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!