174 



(d) 



PRACTICAL MATHEMATICS 



doc 



f j f M 

 Isec x ax = I 



J J cos a; 



J: 



J 



dx 



X . X 



sm 2 - 

 2 2 



9 X J 



sec 2 - dx 



1 - tan'f 



on dividing numerator and denominator by cos 2 -< 



2 



Put ?/ = tan -. Then dy = - sec 2 - dx. 



dy 



Then 

 but 



v/hen 

 when 



Hence 



Isec x dx = 21- - 



A B 



+ = 



i - y z i + 



A(l - y) + B(l + y] = 2 

 y=\ 2B = 2 B = l 



y= - 1 2A =2 A=l 



f t dy t dy 



Isec x dx = I- + |r 



J Jl + 2/ Jl-?/ 



= log e (l + y) - log e (l - y) 



l-tan| 



TC a; 



tan - + tan - 



4 2 | . 7T 



since tan = 1 

 x\ 4 



1 tan - tan - 

 4 2 



98. To integrate tan n x, where n is an even or odd integer. 



dz 



By putting z = tan x, -r- = sec 2 x = 1 + z 2 

 cte 



^ u 

 tan n x dx becomes 



f 2 

 I - 



