To solve the integral of $\cot^2(x) \csc^2(x)$, let's proceed step-by-step:

Problem

$I = \int \cot^2(x) \csc^2(x) \, dx$

Step 1: Use trigonometric identities

Recall the identity: $\csc^2(x) = 1 + \cot^2(x)$ Substituting $\csc^2(x)$ into the integral doesn't simplify directly, so let's expand $\cot^2(x)$ to explore further.

Step 2: Express $\cot^2(x)$ as $(\csc^2(x) - 1)$

We rewrite the integral: $I = \int \cot^2(x) \csc^2(x) \, dx = \int (\csc^2(x) - 1) \csc^2(x) \, dx$ Now expand the product: $I = \int \csc^4(x) \, dx - \int \csc^2(x) \, dx$

Step 3: Solve each term separately

1. Second term: $-\int \csc^2(x) \, dx$: The integral of $\csc^2(x)$ is a standard result: $\int \csc^2(x) \, dx = -\cot(x)$

2. First term: $\int \csc^4(x) \, dx$: For $\int \csc^4(x) \, dx$, we use the reduction formula: $\int \csc^n(x) \, dx = -\frac{\csc^{n-2}(x) \cot(x)}{n-1} + \frac{n-2}{n-1} \int \csc^{n-2}(x) \, dx$ Applying this for $n = 4$: $\int \csc^4(x) \, dx = -\frac{\csc^2(x) \cot(x)}{3} + \frac{2}{3} \int \csc^2(x) \, dx$ Substituting $\int \csc^2(x) \, dx = -\cot(x)$: $\int \csc^4(x) \, dx = -\frac{\csc^2(x) \cot(x)}{3} - \frac{2}{3} \cot(x)$

Step 4: Combine the results

Now substitute back: $I = \left(-\frac{\csc^2(x) \cot(x)}{3} - \frac{2}{3} \cot(x)\right) - (-\cot(x))$ Simplify: $I = -\frac{\csc^2(x) \cot(x)}{3} - \frac{2}{3} \cot(x) + \cot(x)$ $I = -\frac{\csc^2(x) \cot(x)}{3} + \frac{1}{3} \cot(x) + C$

Final Answer:

$\int \cot^2(x) \csc^2(x) \, dx = -\frac{\csc^2(x) \cot(x)}{3} + \frac{1}{3} \cot(x) + C$

Would you like more details or assistance with similar problems?

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Related Questions:

1. What is the reduction formula for integrals of $\csc^n(x)$?
2. How is the identity $\csc^2(x) = 1 + \cot^2(x)$ derived?
3. Can you solve $\int \cot^2(x) \, dx$?
4. What are the standard integral results for $\int \csc^2(x)$ and $\int \cot(x)$?
5. How do you simplify trigonometric integrals effectively?

Tip:

When handling integrals with powers of trigonometric functions, look for identities or reduction formulas to simplify the problem systematically.