integral of cscx^(2/3)(cot^3)x

Question

i know that cot^2x is csc^2(x)-1, but i just don't understand how to solve the cscx^(2/3), any help? i also know that its trig integrals/substitution...

Answer 1

let u = csc(x)
du = -csc(x) cot(x)

cot^2 = csc^2 - 1

and you have

csc^(2/3) (csc^2-1)cot(x)
= csc^(5/3)(csc*cot) - csc^(-1/3)(csc*cot)
= [-u^5/3 + u^(-1/3)] du

all downhill from there