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The integral of $\csc(x) \cot(x)$ can be expressed as:

To compute this integral, we can use the substitution $u = \csc(x)$. Consequently, the derivative $\frac{du}{dx} = -\csc(x) \cot(x)$. This allows us to rewrite the integral as follows:

Next, we can utilize the trigonometric identity $\csc(x) = \frac{1}{\sin(x)}$ to express the integral in another form:

Now, we apply another substitution, letting $u = \sin(x)$. Thus, we have $\frac{du}{dx} = \cos(x)$, which transforms our integral into:

Integrating the expression $-\int \frac{du}{u^2}$ yields:

Thus, we arrive at the final result for the integral: