Discussion of
sec x = ln|sec x + tan x| + C.
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1. Proof
Strategy: The strategy is not obvious. Multiply and divide
by (sec x + tan x); use Substitution.
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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
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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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