Dave's Math Tables: Integral sec(x)

(Math | Calculus | Integrals | Table Of | sec x)

Discussion of sec x = ln|sec x + tan x| + C.

1. Proof

Strategy: The strategy is not obvious.  Multiply and divide by (sec x + tan x); use Substitution.
 

sec x dx =  sec x

sec x + tan x  sec x + tan x

dx

set
  u = sec x + tan x
then we find
  du = (sec x tan x + sec² x) dx

substitute du = (sec x tan x + sec² x) dx, u = sec x + tan x
 

sec x

sec x + tan x  sec x + tan x

dx =

(sec2 x + sec x tan x) dx  sec x + tan x

 

=

du  u

solve integral

= ln |u| + C

substitute back u=sec x + tan x

= ln |sec x + tan x| + C
Q.E.D.