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 Proof: 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.

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