How to integrate csc^4(x)?

To integrate csc^4(x), use the substitution u = cot(x) and partial fractions.

To begin, use the identity csc^2(x) = 1 + cot^2(x) to rewrite csc^4(x) as (1 + cot^2(x))^2 / sin^4(x). Then, substitute u = cot(x) and du = -csc^2(x) dx to get:

∫ csc^4(x) dx = ∫ (1 + u^2)^2 / (1 - u^2)^2 du

Next, use partial fractions to break up the integrand into simpler terms. Write:

(1 + u^2)^2 / (1 - u^2)^2 = A/(1 - u) + B/(1 + u) + C/(1 - u)^2 + D/(1 + u)^2

Solve for A, B, C, and D by multiplying both sides by the common denominator (1 - u)^2 (1 + u)^2 and equating coefficients of like terms. The result is:

A = 1/8, B = -1/8, C = 1/4, D = 1/4

Substitute these values back into the partial fraction decomposition and integrate each term separately:

∫ csc^4(x) dx = ∫ (1/8)/(1 - u) du - ∫ (1/8)/(1 + u) du + ∫ (1/4)/(1 - u)^2 du + ∫ (1/4)/(1 + u)^2 du

= (-1/8) ln|1 - u| + (1/8) ln|1 + u| - (1/2) (1 - u)^-1 - (1/2) (1 + u)^-1 + C

Finally, substitute back u = cot(x) and simplify to get the final answer:

∫ csc^4(x) dx = (-1/8) ln|cot(x) - 1| + (1/8) ln|cot(x) + 1| + (1/2) sin^-2(x) + C