What's the integral of csc^3(x)cot^2(x)?

The integral of csc^3(x)cot^2(x) is -csc(x) - ln|csc(x) + cot(x)| + C.

To solve this integral, we can use the substitution u = csc(x), du/dx = -csc(x)cot(x)dx. Then, we can rewrite the integral as ∫-u^3du.

Using the power rule of integration, we get (-1/4)u^4 + C. Substituting back u = csc(x), we get (-1/4)csc^4(x) + C.

However, we need to express the answer in terms of csc(x) and cot(x). We can use the identity csc^2(x) = cot^2(x) + 1 to rewrite csc^4(x) as (cot^2(x) + 1)^2.

Substituting this into our answer, we get (-1/4)(cot^2(x) + 1)^2 + C. Simplifying this expression, we get -csc(x) - ln|csc(x) + cot(x)| + C, which is our final answer.