Evaluate the integral of csc^3(x) dx.

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

To evaluate the integral of csc^3(x) dx, we can use the substitution u = cot(x) + csc(x). Then, du/dx = -csc^2(x) - cot(x)csc(x) = -u^2, and dx = -du/u^2 - csc(x). Substituting these into the integral, we get:

∫csc^3(x) dx = ∫-du/u^2 = 1/u + C = 1/(cot(x) + csc(x)) + C

To simplify this expression, we can use the identity cot(x) + csc(x) = (cos(x) + 1)/sin(x). Then, we have:

1/(cot(x) + csc(x)) = sin(x)/(cos(x) + 1)

Substituting this back into the integral, we get:

∫csc^3(x) dx = sin(x)/(cos(x) + 1) + C

To further simplify this expression, we can use the identity cos(x) + 1 = 2cos^2(x/2). Then, we have:

sin(x)/(cos(x) + 1) = sin(x)/2cos^2(x/2)

Using the half-angle formula for sine, sin(x) = 2sin(x/2)cos(x/2), we get:

sin(x)/(cos(x) + 1) = sin(x/2)/cos(x/2) = cot(x/2)

Substituting this back into the integral, we get:

∫csc^3(x) dx = cot(x/2) + C

Finally, we can use the identity cot(x/2) = csc(x) + cot(x) to get:

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