Integrate (csc^2 x)/((1 + cot^2 x)(1 + cot x))

To integrate \(\frac{\csc^2 x}{(1 + \cot^2 x)(1 + \cot x)}\), let's first simplify the expression.

### Step 1: Simplify the given expression
We start by recalling the identity:

\[
1 + \cot^2 x = \csc^2 x
\]

So, we can rewrite the denominator as follows:

\[
\frac{\csc^2 x}{(1 + \cot^2 x)(1 + \cot x)} = \frac{\csc^2 x}{\csc^2 x (1 + \cot x)} = \frac{1}{1 + \cot x}
\]

Now the integral becomes:

\[
\int \frac{1}{1 + \cot x} \, dx
\]

### Step 2: Substitute \(\cot x = u\)
Let \(u = \cot x\), so that:

\[
\frac{du}{dx} = -\csc^2 x
\]

This implies \(dx = -\frac{du}{\csc^2 x}\), and since the original integrand involved \(\csc^2 x\), we get:

\[
\int \frac{1}{1 + \cot x} \, dx = -\int \frac{1}{1 + u} \, du
\]

### Step 3: Integrate
The integral of \(\frac{1}{1 + u}\) is \(\ln |1 + u|\). Therefore:

\[
-\int \frac{1}{1 + u} \, du = -\ln |1 + u| + C
\]

### Step 4: Substitute back \(u = \cot x\)
Substituting \(u = \cot x\) back into the result, we get:

\[
-\ln |1 + \cot x| + C
\]

Thus, the final answer is:

\[
\boxed{-\ln |1 + \cot x| + C}
\]

Would you like further details on any of these steps?

Here are some related questions for practice:
1. What is the derivative of \(\cot x\)?
2. How does the substitution method help in integration?
3. Can you prove the identity \(1 + \cot^2 x = \csc^2 x\)?
4. What is the integral of \(\csc^2 x\)?
5. How do you approach integrals involving trigonometric functions and logarithms?

**Tip**: Always check for trigonometric identities to simplify the integrand before proceeding with substitution or integration techniques.

This guide explains how to integrate (csc^2 x)/((1 + cot^2 x)(1 + cot x)) by simplifying the expression and using the substitution method. Step-by-step, the problem involves trigonometric identities and logarithmic integrals, making it suitable for students in Grades 11-12 and college.

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