When approaching the integral, int csc^4(x) cot^6(x) dx, it is helpful to ask about derivatives and integrals of the various functions we see.
d/dx(cscx) = -cscx cotx so perhaps we could split off one of each and rewrite using only cscx. We know that there is a relationship, but it involves squares, not the 3rd and 5th power we have left after separating cscx cotx. We'll keep it in mind if we don't get a better idea.
d/dx(cotx) = -csc^2x. And if we split off a csc^2x, we will have csc^2x remaining and we know that we can rewrite that using cotx, so we'll try that. (with substitution u=cot(x)
(With experience and practice, this reasoning takes place very fast and we know this will work. As students, we have to try something and see if it works.)
int csc^4(x) cot^6(x) dx = int csc^2(x) cot^6(x) csc^2(x) dx
= int (cot^2(x)+1) cot^6(x) csc^2(x) dx
= int(cot^8(x)+cot^6(x)) csc^2x dx
= int (u^8+u^6)(-du) " "(u=cot(x))
= -1/9u^9-1/7u^7 +C
= -1/9cot^9(x)-1/7cot^7(x) +C
