THE INTEGRATION OF tan" x 175 



By division 



I tan" x dx = \z n ~ 2 dz \z n ~* dz+ . . dblazq^l ; 



J J J J Jl + z 2 



when n is even and 



f f f f f z dz 

 Itan" x dx = la"- 2 dz \z n ~* dz . . . T \z dz \ 



J J J J Jl+2 2 



when n is odd. 



(a) When n is an even integer. 

 To integrate tan 6 x 



f f s 6 dz 

 tan 6 x dx = I- r where z = tan x 



f 

 I 

 J 



= fzefe- 



dz 



\z" az+ \az- i- 



J J J l 4- z- 5 



*., l - 



= 5p* T^T ^ g -L v ^ T3n~" ** 



5 3 



= - tan 5 a; - tan 3 x + tan x x 



5 o 



(b) When n is an odd integer. 

 To integrate tan 7 x 



P f 2 7 



Itan 7 x dx= \- - ^ wnere z = tan x 



= \z* dz Is 3 ds+ \z dz \- - 



J J J J2 2 +1 



= - tan 6 x - tan 4 x + - tan 2 x log, sec x 

 642 



99. (a) To integrate r-^ r r 



a sm^ x o cos 2 aj 



f <ir _ j* sec 2 x dx 



Ja sin 2 a? + b cos 2 # Ja tan 2 x + b 



o , where w = tan a; 

 ^ 2 + 6 



-t 



tan -1 <- tan x\ 



... . 

 Vab - ' a 



