∫tan(x−α)tan(x+α)tan2xdx=
1tan(x−α)+1tan(x+α)+1tan2x+C
12tan(x−α)tan(x+α)+tan2x+C
logsec2x+logsecx+α+logsec(x+α)+C
12logsec2x−logsecx+α−logsec(x−α)+C
I=∫tan(x−α)tan(x+α)tan2xdx_ Now tan2x=tan(x+α+x−α)tan2x=tan(x+α)+tan(x−α)1−tan(x+α)tan(x−α)tan2x(1−tan(x+α)tan(x−α)=tan(x+α)+tan(x−α)∴tan(x−α)tan(x+α)tan2x=tan2x−tan(x+α)−tan(x−α)⇒I=∫tan(x−α)tan(x+α)tan2xdx=∫{tan2x−tan(x+α)−tan(x−α)}dx=12log|sec2x|−log|sec(x+α)|−log|sec(x−α)|+C