Question

How do we integrate [(cosx)^2(nx)(sin(nx))]/[a-(cos(nx))] dx?

Answer 1

well, the first thing is to get rid of all those "nx" things. Let u = nx and we have

1/n ∫(cos^2(u)sin(u))/(a-cos(u)) du

now let
v = a-cos(u)
dv = sin(u) du
cos(u) = a-v

1/n ∫(a-v)^2/v dv

Now it's simple.

Answer 2

And what if the question is as follows;
Integrate [(nx)(cosx)^2(sin(nx))]/(A-(cos(nx) )?

After applying the same method as you,(I used both of the substitutions you used)I got

Integrate [(1/n)[u*(cosu)^2 dv]/v] ?
What is the cleverest method to proceed on from here?

Answer 3

tossing in that extra x factor makes it beyond elementary functions, as shown here:

http://www.wolframalpha.com/input/?i=%E2%88%AB(x+cos%5E2(x)sin(x))%2F(a-cos(x))+dx