How to integrate csc^2(x)cot^2(x)?

To integrate csc^2(x)cot^2(x), use the substitution u = cot(x) and simplify.

To integrate csc^2(x)cot^2(x), start by using the identity cot^2(x) = csc^2(x) - 1 to rewrite the integrand as csc^2(x)(csc^2(x) - 1). Then, make the substitution u = cot(x), so that du/dx = -csc^2(x) and dx = -du/csc^2(x). Substituting these into the integral, we get:

∫csc^2(x)(csc^2(x) - 1) dx
= ∫(csc^2(x))^2 dx - ∫csc^2(x) dx
= ∫(1 + cot^2(x))^2 (-du/csc^2(x)) - ∫csc^2(x) dx
= -∫(u^2 + 2u + 1)/csc^4(x) du - ∫csc^2(x) dx
= -∫(u^2 + 2u + 1)(1 + u^2) du - ∫csc^2(x) dx
= -∫(u^4 + 3u^2 + 1 + 2u^3 + 2u + u^2) du - ∫csc^2(x) dx
= -∫u^4 du - 3∫u^2 du - ∫du - 2∫u^3 du - 2∫u du - ∫csc^2(x) dx
= -u^5/5 - u^3 - u - 2u^4/4 - u^2 - ln|csc(x) + cot(x)| + C
= -cot^5(x)/5 - cot^3(x) - cot(x) - cot^4(x)/2 - cot^2(x) - ln|csc(x) + cot(x)| + C

Therefore, the integral of csc^2(x)cot^2(x) is -cot^5(x)/5 - cot^3(x) - cot(x) - cot^4(x)/2 - cot^2(x) - ln|csc(x) + cot(x)|