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#
