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

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

To find the integral of csc^3(x), we can use the substitution u = cot(x) and du = -csc^2(x) dx. Then, we have:

∫csc^3(x) dx = ∫-du/(u^2 + 1) * csc(x)
= -∫du/(u^2 + 1) * csc(x)
= -arctan(u) * csc(x) + ∫arctan(u) * d(csc(x))

To evaluate the second integral, we can use integration by parts with u = arctan(u) and dv = d(csc(x)). Then, we have:

∫arctan(u) * d(csc(x)) = arctan(u) * csc(x) - ∫(1 + u^2) * csc(x) / (1 + u^2) du
= arctan(u) * csc(x) - ∫csc(x) du
= arctan(u) * csc(x) + ln|csc(x) + cot(x)| + C

Substituting back u = cot(x), we get:

∫csc^3(x) dx = -arctan(cot(x)) * csc(x) - ln|csc(x) + cot(x)| + C
= -csc(x) - ln|csc(x) + cot(x)| + C